Cover the plantains with water in a large saucepan and simmer for 15 to 18 minutes, until very tender. Drain well.
Mash with 2 tablespoons butter and warm milk until smooth. Cool 5 minutes, then mix in the beaten eggs.
Cook the Beef:
Heat oil in a skillet over medium-high heat. Brown the beef, breaking it into irregular crumbles, then drain excess fat.
Add onion and pepper and cook 5 minutes. Add garlic, tomatoes, olives, oregano, salt, and black pepper; simmer 10 minutes until thick.
Layer and Bake:
Heat the oven to 375°F (190°C). Grease a 9-by-13-inch dish with the remaining butter. Spread in half the plantain, all the beef, half the cheese, the remaining plantain, and the remaining cheese.
Bake for 25 to 30 minutes until set and browned in patches. Rest 10 minutes before cutting.
Notes
The beef mixture should be moist but not watery before layering.
For a less sweet casserole, replace two ripe plantains with yellow plantains that have only a few black spots.