Place eggplant slices in a colander, sprinkle both sides with kosher salt, and allow to drain in the sink for 30 minutes.
Combine 4 tablespoons olive oil, herbes de Provence, and garlic in a small bowl.
Set an oven rack about 4 inches from the heat source and preheat the oven's broiler.
Pat eggplant dry and press between paper towels to remove as much moisture as possible. Brush slices on both sides with olive oil mixture and place in a single layer on a heavy baking sheet.
Broil eggplant in the preheated oven, turning once after browned, about 10 minutes. Continue cooking on other side until brown and crispy, about 10 minutes.
Grease bottom and sides of a 10- or 12-inch baking dish with 1 1/2 teaspoons olive oil. Sprinkle 2 tablespoons panko over bottom and sides of dish.
Cut eggplant slices in half crosswise on the diagonal and place half the slices in the prepared dish in a single layer. Layer bell peppers on top of eggplant. Arrange goat cheese slices over bell peppers. Season with black pepper and cover with remaining eggplant. Sprinkle with remaining bread crumbs and Parmesan cheese. Drizzle with remaining 1 1/2 teaspoons olive oil. Cover and refrigerate for 30 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Bake in the preheated oven uncovered until gratin is heated through and golden brown on top, 15 to 20 minutes.
Turn oven to broil and place gratin 4 inches from heat source. Broil gratin until breadcrumbs are crispy and brown, 1 to 2 minutes. Allow to cool 5 to 10 minutes before serving.
