One can get flannel patterned shirts that aren’t really flannel. I once had a silk short sleeve shirt that looked like flannel… and I’d sometimes get comments like “aren’t you hot in that?” Sometimes it was a conversation starter with girl, which was a real asset.
My favorite shirts are Sir Pendleton - they’re a very light wool worsted, which makes them warm but really posh, and they are mostly in what you’d think of as flannel-shirt patterns, plaids etc. Several of them in the photo above. Highly recommended if you get any kind of cold weather - they’re like the King of Layers.