Make the pesto: In a food processor, process all pesto ingredients, drizzling olive oil through the top until pesto is smooth. Season pesto with salt and pepper to taste. Set aside.
Make the chicken: Season chicken on both sides with salt and pepper and dredge in flour, shaking off excess. Dip chicken in egg and dredge in flour again until completely coated.
In a large skillet, heat vegetable oil over medium-high heat until pan is hot. Sear chicken 3-4 minutes on each side, flipping once until chicken is golden brown. Transfer to a plate (chicken will not be cooked through)
Saute onion, garlic, and sweet pepper in pan drippings until veggies are just barely tender. Return chicken to pan and spread pesto evenly over each chicken breast.
Finish baking the chicken in the oven at 400 20-25 minutes or until chicken registers 165F. Place a mozzarella slice over each chicken breast and broil 1 minute until cheese is melted. Serve chicken hot and enjoy!