Step 1

Arrange 2 racks to divide the oven into thirds and heat the oven to 425°F. Bring a Dutch oven or large pot of heavily salted water to a boil.

Step 2

Meanwhile, trim and halve 1 pound Brussels sprouts (quarter if large). Place on a rimmed baking sheet (make sure to include any stray leaves). Drizzle with 2 tablespoons olive oil, and season with 1/2 teaspoon kosher salt and several grinds black pepper. Toss to combine, then arrange in a single layer cut-side down.

Step 3

Place the baking sheet on the lower rack. Roast until the sprouts are crispy, tender, and browned in spots, 14 to 16 minutes. Meanwhile, cook the pasta and grate the cheeses.

Step 4

Add 12 ounces cavatappi to the boiling water and cook according to package directions until al dente. Grate the following cheeses on the large holes of a box grater, keeping them separate on the cutting board: 4 ounces Gruyère cheese (about 1 cup), 4 ounces sharp cheddar cheese (about 1 cup), and 1 ounce Parmesan cheese (about 1/2 cup).

Step 5

When the Brussels sprouts are ready, remove the baking sheet from the oven.

Step 6

When the pasta is ready, drain and set aside. Return the pot to the stovetop, add 5 tablespoons unsalted butter, and melt over medium-low heat. Transfer 1 tablespoon into a small heatproof bowl. Add 1/4 cup all-purpose flour, 1/2 teaspoon dry mustard, 1/2 teaspoon garlic powder, and 1/4 teaspoon cayenne pepper to the remaining butter in the pot and cook over medium-low heat, whisking constantly, until fragrant and foaming, about 2 minutes.

Step 7

While whisking constantly, slowly pour in 3 cups whole or 2% milk and whisk until the flour is completely dissolved. Bring just to a full simmer over medium heat, 5 to 7 minutes. Reduce the heat to low and simmer, whisking frequently, until thick enough to coat the back of a spoon, about 4 minutes.

Step 8

Remove the pot from the heat. Add the Gruyère and cheddar cheeses, 1 teaspoon kosher salt, and several grinds black pepper. Whisk until smooth and the cheese is melted. Taste and season with more salt as needed (likely about 1/2 teaspoon kosher). Heat the broiler to high.

Step 9

Add the cavatappi and roasted Brussels sprouts to the pot and stir until well-coated in the sauce. Transfer to the same baking sheet used to roast the Brussels and spread into an even layer.

Step 10

Add 1/2 cup panko bread crumbs and the Parmesan cheese to the reserved bowl of melted butter and stir to combine. Sprinkle the pasta with the bread crumb mixture. Broil until the bread crumbs are golden-brown, 2 to 4 minutes.