1. Preheat broiler with rack 8 inches from heat source. Toast bread on a rimmed baking sheet, stirring once, until crispy, about 2 minutes. Remove from oven; set aside.

2. Scoop out seeds and flesh from each tomato into a bowl; reserve. Arrange tomato halves in a broiler-safe baking dish; season with salt and pepper.

3. In a large skillet, heat 1 tablespoon oil over medium heat. Add pepper flakes and shallot; cook until just tender, about 2 minutes. Increase heat to high; add reserved tomato flesh and juice. Cook, stirring, until reduced by half, about 2 minutes. Add corn and greens; cook, stirring, until vegetables are tender, about 2 minutes. Season with 1/2 teaspoon salt.

4. Fill tomato halves with vegetable mixture, dividing evenly, then sprinkle with toasted bread, drizzle with remaining 1 tablespoon oil, and top with cheese. Broil until cheese is bubbly and golden in spots, about 2 minutes. Serve warm.