4 chayote squash, cut in half
1 tablespoon unsalted butter (such as Kerrygold®)
1 egg, beaten
1/4 cup heavy cream
1/4 cup grated Parmesan cheese
2 tablespoons shredded Cheddar cheese
1 1/4 cups shredded Cheddar cheese
1/2 cup dry bread crumbs
Place the chayote into a large pot and 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 for a minute or two.
Preheat an oven to 375 degrees F (190 degrees C).
Remove the seed and seed membrane from the chayote using a spoon; discard. Scoop out as much of the remaining pulp as possible into a bowl without puncturing the shell. Pat the shell dry with a paper towel and place in a baking dish; set aside. Squeeze the excess water from the reserved pulp. Stir in the butter, egg, heavy cream, Parmesan cheese, and 2 tablespoons Cheddar cheese until well-blended. Fill each of the chayote shells with the pulp/cheese mixture. Sprinkle the remaining 1 1/4 cup Cheddar cheese on top, followed by the bread crumbs.
Bake in the preheated oven until heated through and the cheese has melted, 35 to 45 minutes.