Preheat the oven to 375°F (190°C). Pound the chicken breasts to even thickness using a mallet or rolling pin for uniform cooking.
Season both sides of the chicken breasts with salt and pepper. Heat a skillet over medium-high heat with 1 tablespoon of olive oil. Sear the chicken for 3-4 minutes per side until golden brown, then transfer to a baking dish.
Top each chicken breast with two slices of prosciutto, followed by two slices of mozzarella and three basil leaves. Secure the toppings lightly with toothpicks if necessary.
Drizzle the assembled chicken breasts with remaining olive oil. Place the baking dish in the preheated oven and bake for 20-25 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the cheese is melted and slightly bubbly.
Remove from oven, discard toothpicks if used, and let rest for 2 minutes. Garnish with additional basil if desired. Serve hot, showcasing the melty cheese and crispy prosciutto atop tender chicken.