Preheat oven to 350. Butter a 9 by 5 oven-safe baking dish, set aside. Place oil for frying in a large pan. I like using cast iron.
Begin by prepping the cutlets: butterfly any thick pieces and pound the others slightly so that all the pieces are relatively uniform in thickness. But note, you do not want the cutlets too thin--they will be baked. At this point, you might want to begin heating your frying oil.
Set up a breading station: flour, eggs, breadcrumbs. Season the whisked eggs and flour with salt. Then, dip each cutlet in the flour, followed by egg, followed by a good dredge in the breadcrumbs (making sure the cutlets are fully coated by flipping them in the breadcrumbs at least twice). Fry the cutlets just until they are lightly golden brown on both sides. (Because you'll be baking the cutlets, you do not need to worry about doneness.) Place the fried cutlets in the prepared oven safe dish.
After the cutlets have been lightly fried, add some more oil to coat the pan. Place the sliced mushrooms in the pan, lightly salt them, and sauté until they're cooked through. Add the white wine and chicken broth to the pan, making sure to scrape up all the brown bits from the bottom of the pan. Simmer, until everything has melded together. and the sauce has reduced a bit. Taste and salt as necessary. You can also add a pat of butter if you'd like.
Pour the mushroom sauce over the chicken cutlets and bake for about 15 minutes covered and an additional 15 minutes uncovered. Enjoy.