Butterfly the chicken cutlets then cut each into two pieces for a total of 4 cutlets. Place the cutlets about 3 to 4 inches apart on top of a large sheet of plastic wrap then place another sheet over the top. Using meat mallet or the back of a frying pan, lightly pound the chicken cutlets to about a quarter of an inch thick.
Season the cutlets with salt and pepper on both sides then turn lightly into the flour. Next, coat the cutlets in the egg and finally into the polenta-cheese mixture. Reserve the breaded cutlets on a plate.
Place the chicken stock and milk into medium large saucepan and bring to a boil.
Meanwhile, heat a large nonstick skillet over medium to medium-high heat with a thin layer of vegetable oil. Cook the cutlets in a single layer, in 2 batches if necessary, about 3 or 4 minutes on each side, until the juices run clear and the breading is evenly browned.
Once the stock and milk mixture is at a boil, add the remaining cup of polenta in a slow and steady stream while whisking. Continue whisking until slightly thickened. Add butter and remaining Parmigiano to finish.
To serve, pour some cheesy polenta onto the bottom of each plate the lie the crispy cutlets over it. Top it off with a ladle full of the tomato sauce.