Bring an 8-quart stock pot of water to a boil and generously season with salt. Cook pasta according to package directions until al dente, less 2 minutes. Reserve 1 cup pasta water.
Using a blender, purée beans with 1 cup fresh water until smooth. Set aside. Position rack to the top third of the oven and turn on broiler.
In a large cast-iron skillet over medium heat, heat oil and sauté garlic until softened. Add 1 cup pasta water, bean mixture, 1 teaspoon thyme, and red pepper flakes and season with salt and pepper; bring to a boil and simmer on low for 5 minutes. Mix in 1/2 cup Pecorino and 1/2 cup Parmesan.
Drain pasta and return to pot. Fold in sauce over pasta until just combined. Transfer back to cast-iron skillet. Sprinkle remaining cheese evenly over pasta. Broil until golden brown and bubbling, about 10 minutes. Garnish with remaining thyme leaves and serve immediately.