For the toasts: Put an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Arrange the bread slices in a single layer on the baking sheet and drizzle with olive oil. Bake until light golden, about 10 minutes. Cool for 2 minutes. Rub the warm toasts with the cut side of the garlic. Set aside.
For the topping: In a large skillet, heat the oil over medium-high heat. Add the garlic and red pepper flakes and cook until the garlic is fragrant, about 30 seconds. Add 1/2 of the spinach and stir until it begins to wilt, about 2 to 3 minutes. Add the remaining spinach and cook until wilted, about 2 minutes. Season with salt, to taste. Using tongs, arrange the spinach on top of the toasts. Sprinkle with cheese and bake for 5 to 8 minutes until the cheese is melted and bubbling. Season with salt and cool for 2 minutes.
Transfer the bruschetta to a platter and serve.