Lie-flat Seat Route in FC from East to West Coast

Headed to YVR from East Coast (RIC). Ability to lay flat is ideal for the longest haul, so I am looking for the lie-flat routes between both coasts. The short connections (RIC to nearest hub, and from that destination to YVR) are not a problem, but I'm looking for a good lie-flat seat route for the long haul.

I'm new to United, mostly fly Delta (had the 767 lie-flat ATL-SEA earlier this month which was great, but that is flight schedule doesn't appear with the lie-flat in mid-May). So I am unfamiliar with which United planes/routes have this feature. I don't care if it is First or Business class, just need the seat to go flat.

I'm looking for mid-May Fri/Sat/Sun departure, late Thurs/Fri return.

Thanks for any tips or advice.