Bring a large pot of water to a boil. Add mirliton, and boil until tender, about 45 minutes. Drain, cool, and peel. Cut in half, remove seeds, and cut into cubes.

When the mirliton are almost done, bring another pot of water to a boil. Add shrimp, and cook for about 5 minutes, until pink. Drain shrimp, reserving the water for later. Preheat the oven to 250 degrees F (120 degrees C).

Heat olive oil in a large skillet over medium-high heat. Fry onion and garlic in the oil until soft. Transfer to a 9x13 inch baking dish. Add the shrimp, ham, and cubed mirliton. Beat the egg in a small dish with a tablespoon of the water from the shrimp; pour into the baking dish. Mix in the bread crumbs, parsley, thyme, salt and pepper until evenly blended. If it seems a little dry, add a little more water from the shrimp. You have the choice to freeze, refrigerate, or bake at this point. Sprinkle some more breadcrumbs on top if desired.

Bake for 1 hour and 30 minutes in the preheated oven, or until golden brown on top.