Prepare the Chicken: Pat the chicken pieces dry with paper towels, then season them evenly with 1½ teaspoons of kosher salt and ½ teaspoon of black pepper. Set aside.
Cook the Pancetta: Heat a 10–12-inch oven-safe skillet over medium-high heat. Add the olive oil and let it heat for a few seconds. Stir in the pancetta and cook, stirring frequently, until it is browned and crispy, about 3 minutes. Use a slotted spoon to transfer the pancetta to a plate lined with paper towels.
Sear the Chicken: In the same skillet, add the chicken pieces. Sear them on all sides until they are evenly browned, turning occasionally, about 10 minutes. Once browned, transfer the chicken to a plate. Carefully pour off all but 1 tablespoon of the remaining fat in the skillet.
Build the Sauce: Return the skillet to the heat and add the garlic, anchovy fillets (if using), capers, and red chili flakes. Sauté the mixture for about 1 minute, allowing the flavors to blend. Add the cherry tomatoes, diced tomatoes, and the sprig of basil. Season with the remaining salt and pepper. Cook the mixture, stirring occasionally and breaking down the tomatoes with a spatula, until the sauce begins to thicken slightly, about 10 minutes.
Bake the Chicken: Preheat the oven to 400°F. Return the seared chicken to the skillet, nestling it into the tomato sauce. Place the skillet into the oven and bake uncovered until the chicken is fully cooked and no longer pink, about 30 minutes.
Melt the Cheese: Once the chicken is cooked, remove the skillet from the oven. Scatter the bocconcini or mozzarella pieces over the chicken and sauce. Turn on the broiler and place the skillet under the broiler. Let the cheese bubble and melt, about 1–3 minutes. Watch carefully to avoid burning.
Garnish and Serve: Remove the skillet from the broiler. Sprinkle the crispy pancetta and freshly chopped basil leaves over the top. Serve the dish hot, accompanied by a leafy green salad or green beans with caper vinaigrette, and a side of crusty bread to soak up the flavorful sauce.