Preheat oven to 425°F and position a rack in the upper third of the oven.
Grease a small, rimmed baking sheet liberally with butter. Place 4 slices of bread on the baking sheet, arranging them next to each other to form a large square.
In a medium skillet, heat the olive oil, two turns of the pan, over medium heat. Add scallions, garlic and chile; stir until the mixture softens, about 2 minutes. Scatter the scallion mixture evenly over the bread slices; top with the oregano, salt and pepper. Top with the tomato slices then the remaining bread to form sandwiches that are arranged tightly together like a Sicilian pizza.
In a small bowl, beat the eggs with milk; season. Pour the egg mixture evenly over the bread. Top with the provolone and bake the sandwiches until golden and heated through, 18-20 minutes.
In a medium bowl, toss the greens with lemon juice and olive oil; season. Serve the sandwiches with the greens alongside.