Preheat the oven to 350°F (175°C). Lightly spray a baking sheet with cooking spray.
In a small bowl, whisk the egg with 2 teaspoons water and a pinch of salt to make an egg wash; place the bread crumbs in a second shallow bowl.
Slice each large chicken breast half into about three thin cutlets (you should get roughly 9 cutlets total). If desired, gently pound them to an even thin thickness.
Cut each slice of Swiss cheese in half. Place half a slice of cheese in the center of a chicken cutlet, top with a small spoonful of the chopped cooked broccoli, then fold or roll the chicken to fully enclose the filling and secure with toothpicks. Season the outside lightly with salt.
Dip each stuffed chicken piece into the egg wash, then coat evenly with the bread crumbs. Arrange the pieces on the prepared baking sheet and lightly spray their tops with cooking spray.
Bake for about 25 minutes at 350°F, until the chicken is cooked through and breadcrumbs are golden. Remove toothpicks before serving.
