Adjust oven rack to center position and preheat oven to 400° F. While the oven preheats, set cutlet tray at room temperature.
Line two baking sheets with aluminum foil, and divide the butter and 4 tablespoons of the olive oil between each.
Place the trays into the oven until the butter is melted, about 5 minutes.
Divide the chicken cutlets between the two trays and bake until golden brown on the bottom, about 15 minutes.
Place the cherry tomatoes in an oven proof casserole dish with the remaining olive oil.
Season with salt and pepper.
Turn the cutlets over, return to the oven and cook until golden brown on second side, 10 to 12 minutes longer.
When you turn over the cutlets, place the casserole with the tomatoes in the oven.
Remove the cutlets from the oven, and divide the pesto between each cutlet, then smooth over each cutlet.
Cover the pesto with the mozzarella, then return the cutlets to the oven.
Bake until cheese melts, about 5 minutes.
Serve the cutlets on a platter with the roasted tomatoes alongside.