Heat oven to broil on high.

If using whole breasts (not cutlets), cut them in half to make them thinner. Pat chicken dry and season with salt and pepper.

In a large oven-proof skillet, heat oil and butter over medium-high. Add chicken in a single layer and sear until well browned on both sides and just cooked through, about 10 minutes total. Cook in batches if necessary. Transfer to a large plate.

To the same pan, add garlic, anchovy and red pepper flakes, sauté 1 minute. Stir in tomatoes and basil. Cook until sauce thickens a bit, about 5 minutes.

Return chicken to skillet. overlapping as little as possible. Top each cutlet with some mozzarella and pepperoni. Place skillet in oven and broil until cheese is melted and bubbling and pepperoni is a slightly browned, 2 to 3 minutes (watch carefully to make sure it doesn't burn). Garnish with chopped basil before serving.