In a medium bowl, combine the egg and heavy cream and beat together well. In a separate medium bowl, combine the cracker crumbs, cheese, basil, garlic powder and salt and pepper to taste.
Rinse chicken and pat dry. Dip each piece into the egg wash and then dredge liberally in the cracker crumb mixture. Heat the olive oil in a large skillet over medium high heat. Fry the chicken in the oil for 5 minutes per side, or until both sides are golden brown and crusty. Place chicken in a 9x13 baking dish and cover with foil.
Bake, covered, at 400 degrees F (200 degrees C) for 15 minutes, then remove foil and bake, uncovered, for 15 to 20 more minutes, or until chicken juices run clear.