Arrange the lobster claws on a baking sheet. Broil for 2 to 3 minutes. Turn the claws over. Put the lobster halves, flesh side up, on the baking sheet. Brush the exposed flesh lightly with some of the melted butter. Season lightly with salt and black pepper. Return to the broiler and cook, basting twice with more of the melted butter, for 6 to 8 minutes, or until the lobster meat is nearly cooked through and opaque. Remove from the oven and keep warm. Do not turn off the broiler.
Heat the oil in the top of the double boiler. Add the eggs and cook, stirring constantly, until the eggs begin to thicken and scramble. Add the eggplant, greens, bell pepper, garlic, and thyme. Continue stirring until the eggs are the consistency of a very soft scramble. Stir in the softened butter.
Spoon the scrambled eggs into the cavities of the lobsters. Sprinkle the eggs with the bread crumbs and return the stuffed lobsters to the broiler. Cook until nicely browned and serve.
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