Preheat oven to 350°F.
In a small bowl, beat the large egg with the 2 teaspoons water until combined; set aside as the egg wash.
Place the 3/4 cup whole wheat seasoned bread crumbs in a separate shallow bowl.
On a cutting board, slice each of the 3 (24 oz total) large chicken breast halves into thin cutlets. If pieces are uneven, you may gently pound them to an even thickness.
Sprinkle the single pinch of salt evenly over the chicken cutlets.
Cut each of the 5 reduced-fat Swiss cheese slices in half. Place one half-slice of cheese in the center of each chicken cutlet.
Divide the 2 cups cooked, chopped broccoli evenly among the cutlets and place a small amount on top of the cheese.
Fold or roll each cutlet around the cheese and broccoli to fully enclose the filling; secure each with toothpicks as needed.
Dip each stuffed chicken piece into the egg wash, letting excess drip off, then coat all sides with the bread crumbs, pressing gently so the crumbs adhere.
Lightly spray a baking sheet with oil. Place the coated chicken pieces seam-side down on the sheet and lightly spray the tops with oil.
Bake about 25 minutes, or until the chicken is cooked through and no longer pink in the center (internal temperature 165°F if using a thermometer).
Remove from oven, discard toothpicks, let rest a few minutes, then serve.