Place chayote into a large pot; cover with water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until very tender, 45 to 50 minutes. Drain and allow to steam dry, 1 to 2 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Use a spoon to remove seed and membrane from chayote; discard. Scoop out as much remaining pulp as possible into a large bowl without puncturing the shell. Pat shell dry with a paper towel and place in a baking dish; set aside.
Squeeze out excess water from reserved pulp over the sink. Stir egg, heavy cream, Parmesan cheese, 2 tablespoons Cheddar cheese, and melted butter into squeezed pulp; mix well. Fill chayote shells with pulp-cheese mixture. Sprinkle remaining 1 1/4 cups Cheddar cheese on top, followed by bread crumbs.
Bake in the preheated oven until heated through and cheese has melted, 35 to 45 minutes.