Preheat the broiler.
Peel the skin off two opposite sides of the eggplant, then slice eggplant into 1/4-inch- thick planks. Season with salt and let drain on a kitchen towel for a few minutes.
In a small skillet, heat the EVOO, five turns of the pan, over medium. Add the garlic in a single layer and cook until light golden, about 3 minutes. Remove with a fork, let cool, then chop; reserve. Add the rosemary and crushed red pepper to the EVOO in the skillet.
Pat the eggplant dry; brush with the rosemary-red pepper EVOO. Broil on baking sheet until tender, 2 minutes per side.
Heat a griddle or large nonstick skillet over medium.
Lightly butter one side of the bread slices and sprinkle with the Parmigiano-Reggiano. Build sandwiches, layering bread (buttered side down), a mozzarella slice, grilled eggplant (folded in half), tomato, reserved crispy garlic, basil, anotherminutes mozzarella slice and another bread slice (buttered side up). Grill, turning once, until the bread is golden and the cheese melts, about 5 minutes. Serve with a green salad or tomato soup for dipping.