Place chicken breasts between 2 sheets of heavy plastic on a solid, level surface. Firmly pound with the smooth side of a meat mallet to an even thickness.
Place panko and 1/3 cup flour into separate shallow bowls.
Combine 1 egg, milk, and Dijon mustard in another shallow bowl. Dip each chicken breast into the mixture; coat in the panko mixture. Place breaded chicken breasts on a baking sheet.
Bake in the preheated oven until no longer pink in the center and the juices run clear, 25 to 30 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Reduce oven temperature to 200 degrees F (95 degrees C).
Combine 1 cup flour, whole wheat flour, baking soda, and salt in a bowl.
Mix buttermilk, 1 egg, and 2 tablespoons butter together in another bowl. Whisk gently into the flour mixture. Beat together until batter is smooth.
Preheat a waffle iron according to manufacturer's instructions. Add about 1/2 cup batter into the preheated waffle iron. Cook until waffle is golden brown and the iron stops steaming, about 5 minutes. Transfer waffle to a wire rack set over a baking sheet; place in the oven to keep warm. Repeat with the remaining batter.
Heat 3 tablespoons butter in a heavy saucepan over medium heat. Add 3 tablespoons flour; cook, stirring constantly, until golden brown, 2 to 3 minutes. Whisk in chicken stock slowly. Increase heat to medium-high; bring gravy to a boil. Cook, stirring constantly, until thickened, about 10 minutes. Season with salt and pepper; stir in maple syrup. Reduce heat to low until ready to serve over the baked chicken and waffles.