Place one chicken breast between two pieces of parchment or wax paper. Using a meat mallet or the bottom of a heavy skillet, pound the chicken breast until it is about 1/2-inch thick.
Add the flour to a large, shallow dish. Whisk the eggs and a splash of water or milk in a second large, shallow dish. Stir together the breadcrumbs and grated Parmesan in a third large, shallow dish. Line a plate with parchment paper.
Dredge the chicken in the flour, shaking off any excess, then dip it in the egg wash and finally in the breadcrumbs, pressing to adhere them on all sides. Repeat the breading process with the remaining chicken cutlets.
Line a baking sheet with paper towels.
Place a large stainless steel or nonstick skillet over medium high then generously coat it with oil. (There should be roughly 1/4-inch of oil coating the entire pan.) Once the oil is hot and shimmering, add one or two of the cutlets to the pan and cook, undisturbed, until golden brown, about 3 minutes, then flip it once and continue cooking until it is cooked through and no longer pink.
Transfer the cutlets onto the paper towel-lined plate then repeat the cooking process with the remaining cutlets.
Preheat the oven to broil with the top oven rack at least 6 inches from the broiler. Line a baking sheet with parchment paper (alternately, place a baking rack atop a baking sheet).
Arrange the cutlets on the baking sheet then top them with the marinara sauce and shredded mozzarella cheese. Broil the cutlets until the cheese is melted and bubbling. (This happens for quickly, so keep an eye on them.)
Remove the cutlets from the oven and serve immediately.