The Lasco numbers are S-1016-1 (hot) & S-1016-2 (cold). These are for a Sterling Roman Tub faucet. There are 16 splines on the handle broach.
Thanks for calling out the model numbers; looks exactly like the roman tub filler we have (house was built around 2001). Counted the number of splines and ours also has 16 splines. The cold side is dripping and assume the gaskets need to be replaced (or the entire stem unit). I tried searching for the model number (Sterling #13623) and found the following doc on Kohler's website (which owns Sterling):
https://resources.kohler.com/plumbing/sterling/pdf/97393.pdf . It looks like the one on the lower left image. I was hoping to find a good diagram showing the construction. I ran into a problem when trying to remove the stem unit. The YouTube videos make it look so easy as a DIY project... When trying to remove the stem unit, it seems to be catching on something. Feels like metal on metal. I was hoping to remove it so I could take good pictures/measurement and find replacement parts. I'm wondering if anyone has experience removing this type of stem unit. e.g., do you need a special tool, do you have to remove some other part, etc. I've attached a bunch of pictures. The later images show how far I was able to pull it up. Any help is greatly appreciated. Thanks!