HF in first person

I can't think of any written in first person, but Elizabeth Chadwick has written several in third person. Her books are about historical figures in medieval England, like William Marshal. She has such a grasp on the time period.

Scratch that, I just thought of one. Lady Macbeth by Susan Fraser King, written about the historical Lady Macbeth, is written in first person. There's some Gaelic in there, for authenticity's sake.

The Town House by Norah Lofts.
There's no theeing and thouing or other archaisms, just avoidance of anything that stands out as modern. (There are occasional bits of Suffolk dialect for a few characters.)
She achieves a sense that these are medieval people not modern people by getting right into their heads and having an enormous amount of knowledge about their lives and beliefs.

It's on Amazon as an ebook so you can read the sample for free.

Philippa Greggory writes in first person a lot, so does Bernard Cornwell. I can't say whether either writes in the specific time period you're interested in (as I haven't read all of their books), but I feel like I wasn't drawn out of the narrative by the first person language. And this is more Renaissance, but both of Elizabeth Loupas's novels are in first person, and I felt she struck a good balance between being understandable and not being "modern" in tone.

Stay tuned - if I ever get off my bum and agent "The Ax and the Vase" and get it sold, that is in first person. The period is what most call the "Dark Ages" (a poor term, but understandable at least) but it could be of interest as well.