Preheat oven to 350 degrees F. Prepare rice mixes according to package directions. Spread cooked rice in the bottom of an ungreased 3-quart baking dish.
Meanwhile, for sauce, in a medium saucepan, cook mushrooms, green onions, and garlic in hot butter over medium heat until tender. Stir in the 2 tablespoons flour. Gradually stir in half-and-half. Cook and stir until thickened and bubbly. Stir in shredded cheese until melted. If desired, stir in sherry. Spoon sauce over rice; cover and keep warm.
Starting from the thickest side of each chicken breast half, make a horizontal slit to, but not through, the other side. Wrap each stick of cheese in a slice of ham and insert into a slit. Secure with wooden toothpicks. Sprinkle chicken with salt and pepper.
Place the 1/3 cup flour in a shallow dish. In a second shallow dish, combine eggs and water. Place panko in a third shallow dish. Dip chicken in flour, shaking off excess; dip in egg, then in panko, turning to coat.
In an extra-large skillet, heat 2 tablespoons of the oil over medium heat. Cook chicken, half at a time, in hot oil about 4 minutes or until brown on both sides, adding the remaining 2 tablespoons oil with the second batch. Remove toothpicks. Place chicken on top of sauce.
Bake, covered, for 40 to 45 minutes or until chicken is no longer pink (170 degrees F). Makes 6 servings.