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 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 transfer back to cast-iron skillet.

fold in sauce over pasta until just combined.

sprinkle remaining cheese evenly over pasta.

broil until golden brown and bubbling, about 10 minutes.

serve immediately.