Prep the Mirlitons: Peel, core, and slice the mirlitons. Boil them in a pot of water for 5-7 minutes until slightly tender, then drain.
Make the Sauce: In a skillet, melt butter over medium heat. Add the onions and garlic, cooking until soft. Stir in the heavy cream, chicken broth, nutmeg, thyme, salt, and pepper. Let it simmer for 5 minutes until the sauce thickens.
Assemble the Casserole: Preheat the oven to 350°F (175°C). In a baking dish, layer the cooked mirlitons, then pour the creamy sauce over the top. Sprinkle with cheddar cheese, Parmesan, and breadcrumbs.
Bake: Bake for 20-25 minutes, or until the top is golden and bubbly.
Serve: Let it cool for a few minutes before serving.