Bring a large pot of salted water to a rolling boil.
Add the broccoli florets and cook for 2 minutes until bright green and just tender, then remove with a slotted spoon and plunge into an ice bath or run under cold water to stop cooking.
In the same boiling water, add the pasta shells and cook according to package directions until al dente; drain and return the pasta to the pot.
While the pasta cooks, melt the butter in a saucepan over medium heat.
Whisk in the flour to form a roux and cook, whisking, until golden and nutty-smelling, about 2 minutes.
Combine the milk and heavy cream, then slowly stream about 1/4 cup at a time into the roux while whisking constantly until fully incorporated and the sauce begins to thicken.
Stir in the cream cheese until melted and smooth, then remove the pan from heat.
Add about 12 ounces (¾ of the grated cheddar) to the warm sauce in small handfuls, stirring gently until melted before adding more; do not raise the heat to avoid graininess.
Taste and season the cheese sauce with kosher salt and black pepper as needed.
Pour the cheese sauce over the drained pasta in the pot, add the blanched broccoli, and stir to combine.
Serve immediately, stirring in the remaining grated cheddar as you plate for extra meltiness if desired.
