Step 1

Arrange a rack in the middle of the oven and heat the oven to 400F.

Step 2

Bring a large pot or Dutch oven of heavily salted water to a boil. Meanwhile, trim the stalks from 1 pound broccoli, then cut the florets into 1/2-inch pieces. Cut 2 ounces cream cheese into 1/4-inch cubes. Grate 8 ounces sharp cheddar cheese on the largest holes of a box grater (about 2 cups). Finely grate 2 ounces Parmesan cheese (about 1 cup). Mince 2 garlic cloves.

Step 3

Add 1 pound dried cavatappi pasta to the boiling water and cook 2 minutes less than al dente, about 6 minutes or according to the package instructions. In the final minute of cooking, add the broccoli. It will turn bright green. Drain the pasta and broccoli.

Step 4

Melt 4 tablespoons unsalted butter in the now-empty pot over medium heat. Add the garlic and cook, stirring occasionally until fragrant, about 30 seconds. Add 1/4 cup all-purpose flour and whisk to combine. The mixture will look like wet sand. Cook, stirring constantly, until the mixture begins to dry out,1 to 2 minutes.

Step 5

Slowly whisk in 4 cups whole or 2% milk and continue whisking until smooth. Bring to a boil over high heat. Reduce the heat to medium-low and simmer, stirring occasionally, until thickened enough to coat the back of a spoon, 4 to 7 minutes.

Step 6

Remove the pot from the heat. Add the cream cheese, cheddar, half of the Parmesan, and 1 teaspoon kosher salt. Stir until the cheese is melted and the sauce is smooth. Add the cavatappi and broccoli and stir to combine.

Step 7

Transfer the mixture to a 9x13-inch baking dish and spread into an even layer. Sprinkle with the remaining Parmesan cheese. Bake until lightly browned on top and the sauce is bubbling, 15 to 20 minutes.