With a meat mallet or a rolling pin, pound the chicken cutlets between sheets of plastic wrap to 1/8-inch thickness. Season the cutlets with salt and pepper and press the seasonings into the meat with your fingers.
Make the coating: Combine the bread crumbs, cheese, parsley, salt, and pepper in a small bowl. Work in the olive oil by hand, then spread the mixture on a dinner plate. Spread the flour on another dinner plate. Break the eggs into a shallow bowl and beat lightly. Dip the cutlets in the flour, coating both sides and shaking off the excess. Then dip in the egg, letting any excess drip back into the bowl. Finally, coat the cutlets on both sides with the seasoned bread crumbs, pressing them into place. As each cutlet is coated, place it on a tray. Cover the cutlets and refrigerate until you are ready to fry them.
Heat a large skillet over high heat until very hot. Add olive oil to a depth of 1/4-inch. When the oil is almost smoking, add as many cutlets as the skillet will hold comfortably; do not crowd the pan. Cook until the cutlets are golden on the bottom, about 1 minute. Turn and cook on the second side about 30 seconds to 1 minute longer. With tongs, lift the cutlets as they are done, allowing any excess oil to drain back into the skillet, and transfer to cookie sheet lined with paper bag to drain excess oil. Repeat with the remaining cutlets.
When ready to serve, place chicken cutlets on cookie sheet to finish cooking, about 2 minutes.
Place the arugula in a large bowl. Drizzle with enough extra-virgin olive oil to coat the leaves lightly. Add a squeeze of lemon and some salt and pepper. Toss, taste, and adjust the seasoning.
Divide the cutlets among 4 plates, placing them in the center. Spoon a little Salsa Rosa on each side of the cutlets. Mound the arugula salad on top. With a vegetable peeler, shave a little Parmesan on top of the salad.
Michael's Notes: If you like my version of Chicken Milanese, and I know you will, try the same method on pounded veal, turkey breasts or on very thin swordfish steaks.
Mama's Salsa Rosa:
Preheat oven to 450 degrees F.
Line a baking sheet with aluminum foil. Brush the bell peppers lightly with olive oil and place on the baking sheet. Bake, turning every 10 to 15 minutes, until the peppers are blistered all over, about 30 minutes total. Transfer the peppers to a brown paper bag, and close tightly so they steam as they cool. You can roast the peppers under the broiler or over a gas flame, if you prefer, but I like to do them in a hot oven. It takes a little longer, but you don't have to monitor them as closely.
While peppers are steaming, heat the 2 tablespoons olive oil in a small skillet over high heat. Add the whole chiles, lower the heat to moderate, and cook, turning occasionally, until the chiles are softened, lightly browned, and blistered on all sides, about 8 to 10 minutes. Remove the skillet from the heat and let the chilies cool in the oil for several minutes. When cool enough to handle, remove the stems.
Add more oil to the skillet, if needed, to make 2 tablespoons. Reheat the oil over moderate heat and add the garlic. Cook until lightly browned, about 30 seconds, then add the oregano and saute briefly to release its fragrance. Add the tomato puree and salt, bring to a simmer, and simmer for about 5 minutes to thicken slightly.
Remove peppers from bag, peel and remove the stems, seeds, and ribs; avoid the temptation to rinse the peppers, which would wash away flavor. You should have about 4 cups.
Combine the red peppers, serrano chiles, and tomato sauce in a blender and puree until smooth. Add the vinegar and the black pepper to taste and puree again. Taste and adjust the seasoning. Refrigerate for up to 5 days or freeze for up to 4 months.