Heat the oil in a skillet over medium-high. Season the chicken with pepper (the prosciutto will provide the salt). Sauté the chicken until golden brown on both sides, but not cooked through, about 2 minutes per side. Transfer chicken to a plate.
Preheat the oven to 375 degrees F. Line a baking pan with parchment paper. Combine the tomato sauce, chicken stock and pesto in a small pot and set aside. Set a piece of the cheese on top of each piece of chicken. Wrap each chicken breast with a piece of prosciutto and set in the baking pan, cheese-side up. Bake until the cheese is melted and the chicken is cooked through, about 15 minutes. While the chicken cooks, loosely cover the tomato sauce mixture and simmer 5 minutes.
Make a pool of the sauce on each of 4 plates. Set a piece of chicken on each plate and serve.