Preheat oven to 400˚F. Cover a sheet pan with aluminum foil.
Slice the breasts in half horizontally. If the pieces are still thick, put them between two pieces of parchment paper and pound them with a meat mallet or rolling pin to 1/4-inch thickness.
Season chicken breasts generously with salt and pepper. Place the flour on a dish or wide bowl and lightly coat each seasoned chicken breast with the flour, shaking to remove the excess.
Heat a large skillet over medium high and melt 2 tbsp of the butter and 2 tbsp of the oil. Add half the chicken to the skillet and cook undisturbed, turning once, until golden brown outside and cooked through, about 3 minutes per side. When ready to turn, the chicken will release easily from the pan with no tugging. Transfer to chicken to sheet pan and repeat with the remaining butter, oil and chicken.
Assemble the saltimbocca by covering each chicken breast with a slice of Prosciutto. Next, place one or two whole basil leaves on top of the Prosciutto and then lay two slices of fresh mozzarella overtop, covering the basil leaves.
Place the sheet pan into the preheated oven and bake until cheese is melted and bubbling. This takes about about 5 minutes. Transfer from sheet pan to serving platter, sprinkle with kosher salt and pepper and serve!