Brown the ground turkey on medium heat in large saute pan and season with salt and pepper. Use a wooden spoon to break the meat up into small pieces.
Add the chopped onions, garlic, salt, pepper, tomato and cilantro and continue cooking on a low heat. Add alcaparrado and about 2 tbsp. of the brine, cumin, oregano, bay leaves, and more salt if needed. Add tomato sauce and 1/4 cup of water and mix well. Reduce heat to low and simmer, covered, about 15 minutes.
Preheat oven to 350 degrees F. Lightly spray a 9x9 baking dish with oil. Heat a large nonstick saute pan over medium heat. When hot, spray with oil to lightly coat the bottom. Cook plantains for 2-3 minutes on each side or until golden; set aside.
Assemble lasagna by layering the plantains, then the cooked turkey picadillo, then about 1/4 cup of cheese, and repeat. You want to finish with a layer of plantains. Beat the egg and egg whites with 2 tbsp. of milk and pour over the pastelón to help bind everything, letting it sit for a minute or two to soak in. Top off with remaining cheese. Cover tight with foil and bake for 30-35 minutes, or until completely heated through. Cut into 9 equal pieces and serve.