Preheat the oven to 425°F (if baking) or preheat the air fryer to 400°F and lightly spray a sheet pan or air fryer basket with oil.
Place the chopped broccoli in a microwave-safe bowl with 1 tablespoon water, cover, and microwave 1 minute until softened; drain and season with 1/4 teaspoon kosher salt.
In a small bowl whisk together the egg, 2 teaspoons water, and a pinch of salt to make an egg wash; place the breadcrumbs in a separate shallow bowl.
If needed, pound each chicken cutlet to about 1/4-inch thickness between sheets of wax paper or plastic wrap.
Season both sides of each cutlet with the remaining 1/2 teaspoon kosher salt.
Place a half slice of cheddar in the center of each cutlet and top with about 2 tablespoons of the cooked broccoli.
Roll each cutlet around the filling, securing the ends with toothpicks if needed so the cheese is fully enclosed.
Dip each roll in the egg wash, then coat evenly with the seasoned breadcrumbs and transfer to the prepared sheet pan (or work surface if using the air fryer).
Lightly spray both sides of the breaded rolls with olive oil spray.
For oven: bake on the sheet pan at 425°F about 25 minutes until cooked through. For air fryer: place in the air fryer basket in a single layer (cook in batches if needed) and cook at 400°F for 14–16 minutes, turning halfway, until cooked through.
Remove toothpicks before serving and serve warm.