Preheat oven with broiler setting, and position a rack several inches below.
Sprinkle the pounded chicken breasts generously with salt and pepper. Set up three low shallow bowls for dredging. The first bowl will have flour. Next, combine the 2 eggs with a Tbsp of water and beat. Last, combine the parmesan, breadcrumbs, red pepper flakes and oregano.
Start by dredging each breast through the flour, shaking off excess as you go. Then dip both sides into the egg wash, and last, coat the chicken with the breadcrumb mixture. Place the coated chicken breasts on a platter until ready to cook. This step can be done in advance.
Heat a large cast iron or non-stick skillet over medium-high heat, and melt 2 Tbsp each of EVOO and butter. Add 2-3 chicken breasts at a time, depending on the size of the chicken. You want to sear the crumbs to golden brown and just barely cook the chicken through - about 5 to 6 minutes per side. Repeat the process with the remaining chicken breasts, adding more oil and butter to the skillet as you go. Place the fully cooked chicken into a 9x13 baking dish.
Heat the Rao's sauce over low heat, and pour over the chicken breasts to cover. Split the mozzarella (or burrata) over the top and sprinkle with extra grated parmesan. Broil until the cheese is melted and golden brown in spots. Garnish with freshly torn basil.