I had maybe a second between this bird unexpectedly taking off, me raising the camera, and my good ol' D500 and 500mm PF focusing perfectly. No way did I think I got the shot.
Move the focus point to the near eye, and in this case since the back eye is farther away from the camera, a smaller aperture would help to get both eyes in focus. Hope that helps. :)
I have no problem with the background. The only issue to me is that the point of her nose is in sharp focus, but the eyes are slightly out of focus. Nice expression and it has a nice mood.