What size beam should be used for a 15′ patio cover span

I'm looking into building a covered patio and am having a bit of difficulty finding the proper beam size required for the spans between posts.

The cover will consist of either a ledger board attached to the exterior wall (pending structural review) OR beams on the new patio slab leading to a 15' span away from the house to 6×6 posts on galvanized anchor brackets at the other end of the slab. The patio will be 30ft. wide and 15ft. deep, but this can be adjusted if it significantly cuts down on lumber to do 14ft spans or so. So, either 6 posts in a 3×2 grid or 3 posts and a ledger board.

I live in North Texas where snow is extremely rare (maybe once a year with a couple inches). Will match roof pitch to my home (looks like a 3:12 or 4:12). I can add knee braces. Post height will be 8ft.

Wood species has not been chosen, but pressure-treated pine would be nice as I can easily source posts with that too.

This link seems to indicate I would need 3.5×11.25 #2 PT beams. I'm not sure if I'm using the right tables or not, hence my question here. It seems to me that two 2x12s with a 1/2 plywood along the grain sandwiched in-between would be more than suitable judging by some of the new construction I've seen around here. That sound right?

I would really like to figure out what would be required now so that I can figure out a rough estimate of the project cost and decide if I want to proceed with it. Otherwise, I would have just hired a structural engineer to define them for me already.

Here's an example of what I'm looking at creating. Only the ledger would attach to a two-story wall. I know a 4×6 beam likely wouldn't be sufficient and I'll have more stout 2×10 joists at the recommended spacing for the span (24" on center).