Pour beaten eggs into a shallow dish or bowl. In another shallow dish or bowl, mix together the grated Parmesan cheese and bread crumbs. Dip chicken breasts into beaten egg, then into bread crumb mixture to coat.
In a large skillet, heat oil over medium high heat. Add coated chicken and saute for about 8 to 10 minutes each side, or until chicken is cooked through and juices run clear.
Pour tomato sauce into a lightly greased 9x13 inch baking dish. Add chicken, then place a slice of Monterey Jack cheese over each breast, and bake in the preheated oven for 20 minutes or until cheese is completely melted.