Heat broiler with rack 8 inches from heat source.
Toast bread on a rimmed baking sheet, tossing once, until crisp, about 2 minutes.
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.
In a large skillet, heat 1 tablespoon oil over medium.
Add red pepper flakes and shallot; cook, stirring frequently, 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 salt.
Fill tomato halves with vegetable mixture, dividing evenly, then sprinkle each half with toasted bread, drizzle with remaining tablespoon oil, and top with mozzarella.
Broil until cheese is bubbly and golden in spots, about 2 minutes. Serve immediately.