Preheat oven to 400 degrees F (200 degrees C). Grease a 9x13-inch baking dish with butter.
Slice eggplant into 1/4-inch thick slices lengthwise; discard the end slices with skin. Lay the eggplant slices on paper towels and sprinkle both sides generously with sea salt. Let slices drain for 30 minutes; rinse and pat dry with paper towels.
Place flour, eggs, and bread crumbs in separate shallow bowls. Dip each eggplant slice in flour and shake off excess; dip into egg and then into bread crumbs.
Heat vegetable oil in a large skillet over medium heat and fry the slices until golden brown, about 5 minutes per side. Drain on paper towels, place on baking sheets, and refrigerate eggplant for 10 minutes.
Mix ricotta cheese, Romano cheese, and garlic in a bowl; season to taste with salt and black pepper. Place a cooled eggplant slice onto a work surface and mound 1/3 to 1/2 cup of cheese mixture in the center; roll up slice and place seam side down into the prepared baking sheet. Repeat with remaining slices and cheese filling.
Bake rolls in the preheated oven until filling is hot, about 25 minutes.