I have recessed (can) lights in our vaulted ceiling living room.
I wish we had done more! We're also going to put hanging lights on the beams. Sorry no picture right now - but I've seen it done so many times. It's not strange at all.
However, for a kitchen you will want a secondary kind of lighting as can lights create a lot of shadows and you might not see what you are chopping! :)

Here are mine:

We live in CA and are subject to Title24 requirements for fluorescent lighting so this was quite a challenge. We were limited in the depth so could not do fluorescent cans--the pendants are fluorescent and we were very limited on the choice. At some pt. I'll probably change those out.

The trim for the cans are adjustable so lights can aim downward and not to the side. They work very well.

Thanks for the photo. It really helps envision it. I have a pendant over the sink, which should help make it bright enough. Maybe I can talk him in to some under cabinet lighting, too!