As a half day trip from Madrid, what, in your opinion, is the best way to see both sites? Thanks.

Tom,
Having just come today from El Escorial (now in my own "back yard" so to speak), I would not. recommend at all that you visit the Valley of Fallen, now renamed, thankfully, Valle de Cuelgamuros, (although the entrance to this cold and gloomy site still says "Valle de los Caídos¨). Generalísmo Franco is no longer interred there, but it is still considered very much a pro-Francoism site, a monumento franquista, and still controversial and frankly, gives me the chills, due to its history.
It sits 20 minutes (13.6 kilometers) from the center of San Lorenzo del Escorial on the M-600. I just past there this afternoon, coming home from Segovia (I live within a half hour's drive), and while it is within my path on a weekly basis. going back and forth to El Escorial and beyond (Segovia, Avila) it never occurs to me to stop and tour this site. But I have been.... once was enough. Just keep in mind its history....
Just one opinion, however.
Bus 660 (or 664?) will take you from the bus station at San Lorenzo de El Escorial out to the site, but the bus doesn't run frequently. Otherwise, one must drive or take a taxi.

I went with a half day tour, and that worked well for me. I had some very interesting conversations with other people on the tour, many of whom were Spaniards who had gone specifically to spit on Franco's grave, which was then ln the basilica. I think his remains may have subsequently been relocated to a different place in that area.

I'm sure you realize that there are any number of complicated ethical questions about how or whether to visit sites built by slave labor, once viewed as a tribute to Franco; I won't pretend to have an answer for you on that front!

(I posted at the same time as Maribel.)

Yes, indeed, there are many, many complicated ethical questions, especially if one reads up on the history of the building of this monument and the issue of the slave labor (the losing side in the Guerra Civil) involved in its construction. Franco's remains were removed in 2019, so no more spitting on his grave.
The remains of Primo de Rivera, the leader of the Falangist movement have also been removed. But it is still revered among the remaining "Falangistas".

When our US friends come to see us, I don't include it on our usual sightseeing tour, San Lorenzo de El Escorial, Segovia, San Ildefonso de La Granja, Avila.... because of the ethical questions involved, its Fascist history and because I find it just very gloomy, solemn, utterly cold and rather creepy.
But again, I have indeed visited it once, simply out of curiosity, so there's that...!

But if you come to our house to visit, I'll take you there, drop you off, and pick you up! As the avid student of history that you are! Then we can go to Luz de Lumbre in San Lorenzo de El Escorial for a delicious grilled steak of pizza to warm up!

https://europeanconservative.com/articles/news/spain-launches-controversial-overhaul-of-valley-of-the-fallen/#:~:text=The%20Valley%20of%20the%20Fallen%20is%20a %20vast%20monumental%20complex,say%20it%20glorifie s%20Franco's%20regime.

I have many fond memories of touring Escorial years ago which is amazing, but decided to bypass the Valley of the Fallen when Franco was still interred there. It way on to lovely Segovia. We had a car but I am sure there might be other options from Madrid to El Escorial.

Again, Franco's remains were removed October 29, 2019.

https://www.patrimonionacional.es/en/visita/cuelgamuros-valley-0

For anyone reading, the Monasterio de El Escorial is closed on Mondays!
https://www.patrimonionacional.es/en/visita/royal-site-san-lorenzo-de-el-escorial

From Madrid to San Lorenzo de El Escorial, the current public transportation option is to take the Bus 664 from the Intercambiador de Moncloa (the Moncloa intermodal bus station which I use almost daily, since it's my local commuter bus station as I live between my suburb and San Lorenzo de El Escorial),
which offers buses directly to the San Lorenzo de El Escorial bus station within walking distance of the monument.
They leave regularly.as in every 15 minutes or so, from Monday to Friday.
Here's the Monday-Friday bus schedule,
https://www.alsa.es/documents/216436...=1591198429842

Nice dining in the area
https://luzdelumbre.es
https://www.vestataberna.com
https://www.cavaaltarestaurante.es

My husband and I were in Madrid in the winter of 1981 and went El Escorial and Valle de los Caidos as a day trip. The former was magnificent and we greatly enjoyed the visit.

Looking back I don’t believe we had any idea what the latter really was. This was long before internet/Google searches, etc. We did have a guidebook, but I haven’t a clue if it did truly give the facts. All I remember is that it was a cold and dreary day and we were in a hurry to get back to our hotel in Madrid.

Thanks for all the replies regarding El Escorial and Valle de Cuelgamuros. Maribel, I recognize the latter's history is "complicated" at best, but like you were, I am curious about it. The missions here in California also have a complicated past (involving forced Native American labor), but I still go occasionally to one I've not visited, realizing its past while there. I've been to two concentration camps (one in Germany and one in Austria) and recognize that much of history has an ugly side to it. I am on the fence, but knowing my curiosity, we will probably go. I appreciate your offer, but don't want to impose on you ... and don't forget dinner. I better start looking, it's only five months away.

I have made the decision to pass on visiting Valle de Cuelgamuros. Maribel, thanks for the bus schedule.

I visited it some years ago. I visit sites for historical reasons and of interest to see things, not because I support the person there at all, in many cases, if that's the issue. For example, many people visit concentration camps in Germany, Poland etc which I've never had the slightest interest in doing. It's not like I am ignorant of what happened there and I've seen many many photos of them, etc and visited several HOlocaust museums, I don't need to visit the sites to know what that is about. I probably read and know more history of WWII than a lot of people, actually as it's a particular interest of mine. I've visited numerous Napoleon sites not because I think he was wonderful, au contraire, but because of my interest in history of his period and the Prussian war. There are plenty of sites in the US built with slave labor people visit. There is nothing wrong with visiting them if the guide has a good discussion of the issues. In fact, I just read a book about that which was well regarded (How The Word is Passed by Clint Smith) and Jefferson's Monticello was one example of where the guides are doing a good job. I toured a Communist built workers' town outside Krakow by a local guide who lived through that period (built for steelworkers) and he was fascinating. They had torn down the Lenin statue, though. I would have been interested in seeing it for historical reasons but obviously the local feelings didn't want it around even for history.

I did use a bus to go to Escorial from the Moncloa station, as I recall.

I didn't visit that Valley of the Fallen as I was a DIYer.

I took the bus to El Escorial in 2018 and thoroughly enjoyed it. I looked into making a two for one trip, but getting over to the other site was not straightforward.