Use a sharp, thin knife to slice the chicken breasts in half lengthwise. Season both sides with kosher salt and freshly ground black pepper. Heat a large oven-proof skillet over medium high heat and add the olive oil and butter. Once the butter has melted into the olive oil, add the chicken breasts to the pan, being careful not to crowd. Cook on each side until lightly browned, about 3-4 minutes each.
Slather the tops of each chicken breast with the basil pesto, about 1-2 tablespoons per chicken breast. Top each chicken breast with a slice of mozzarella and a few slices of tomato. Put the skillet in the oven and cook for 10-12 minutes or until the chicken reaches an internal temperature of 165 degrees. F. Remove from the oven and garnish with fresh basil and a drizzle of balsamic glaze.