Heat the oven to 375°F. If needed, lightly pound the chicken thighs so they have an even thickness, then season both sides with the divided kosher salt and freshly ground black pepper.

Pat the fresh mozzarella dry. Place one prosciutto slice and one mozzarella stick or thick slice in the center of each chicken thigh, adding a little chopped basil if desired; roll each thigh tightly around the filling and secure it with a toothpick.

Heat 2 tablespoons olive oil in a large ovenproof skillet over medium-high heat. Brown the stuffed thighs for about 2–3 minutes per side, beginning seam side down, then transfer them temporarily to a plate.

Add 1 tablespoon olive oil to the skillet if needed. Cook the chopped onion for 3–4 minutes, add the minced garlic, and cook for about 30 seconds.

Add the tomatoes, dried oregano, red pepper flakes if using, remaining kosher salt, black pepper, and dry white wine or water. Simmer for 5 minutes, breaking up whole tomatoes if necessary.

Return the chicken to the skillet and spoon some tomato sauce over the rolls. Transfer the skillet to the oven and bake for 20–25 minutes, until the thickest part of the chicken reaches 165°F and the mozzarella is melted.

Let the chicken rest for 5 minutes, then carefully remove every toothpick. Spoon the tomato sauce over the stuffed thighs and serve while hot.