Preheat the oven to 475°F and position a rack in the upper-middle slot.
Heat 2 tablespoons of the olive oil in a medium saucepan over medium heat; add the chopped onion and cook until softened, about 3 minutes.
Add the minced garlic to the pan and cook until fragrant, about 30 seconds, then stir in the can of diced tomatoes and simmer until the sauce is slightly thickened, about 5 minutes.
Remove the sauce from the heat and stir in half of the chopped basil. Pour half of the sauce into the bottom of a 9x13-inch baking dish.
In a medium bowl, combine the shredded mozzarella and grated Parmesan; reserve 1/2 cup of this cheese mixture for topping and stir the remaining basil into the rest of the cheese.
Pat the chicken cutlets dry and season both sides with salt and pepper. Place each cutlet on a work surface and top with the basil-cheese mixture, leaving a 1-inch border at the bottom edge.
Roll each cutlet tightly from the top toward the bottom edge and arrange the rolls seam-side down in the prepared baking dish on top of the sauce.
Toss the remaining 2 tablespoons of olive oil with the panko bread crumbs in a small bowl. Sprinkle the reserved 1/2 cup cheese over the rolls, spoon the remaining sauce over them, then evenly sprinkle the breadcrumb mixture on top.
Bake until the chicken is cooked through and the crumbs are golden, about 15 to 20 minutes; if thicker breasts are used, bake an additional 10–15 minutes until the internal temperature reaches 165°F. If crumbs brown too quickly, tent loosely with foil for the remaining time.
Let rest briefly before serving.
