After thinning the chicken breasts with a mallet or rolling pin, season them all over with salt and pepper.
Set up your dredging station: In a medium pan or plate, mix together the flour with 1 tsp salt and ½ tsp pepper. In a 2nd pan or medium bowl, mix together the lightly beaten eggs with the milk (or water). Finally, in another medium pan, or plate, mix together the bread crumbs and Parmesan cheese with the dried herbs.
One cutlet at a time, first, dredge the chicken in the flour until fully coated. Next, dip in the egg bath to coat, shaking off excess. Next, press into the bread crumbs until fully coated. Place on a plate or platter, and repeat with remaining cutlets.
Heat the oil in a large skillet over medium heat until shimmering. Add the butter and swirl until melted. Working in batches (don't overcrowd the pan), place the chicken cutlets into the pan. Cook until nicely browned on the bottom, about 3 to 4 minutes. Flip over and cook until the other side is also browned and the cutlet is fully cooked, another 3 to 4 minutes. Place on a plate lined with paper towels and continue cooking remaining cutlets, adding more butter if necessary.
Serve at once with vodka sauce, if desired.