Bring a large pot of salted water to a rolling boil.
Add the broccoli florets to the boiling water and cook for 2 minutes, then remove with a slotted spoon and plunge into an ice bath or run under cold water to stop cooking; set aside.
Add the pasta shells to the same boiling water and cook according to package directions until al dente; drain in a colander and return the pasta to the pot.
While the pasta cooks, melt the butter in a saucepan over medium heat.
Whisk the flour into the melted butter to form a roux and cook, whisking, until it smells nutty and is lightly golden, about 2 minutes.
In a measuring cup combine the milk and heavy cream. Gradually stream about 1/4 cup at a time into the roux while whisking constantly until fully incorporated and slightly thickened.
Stir in the cream cheese until melted and smooth, then remove the saucepan from the heat.
Add about 12 ounces (¾ of the grated cheddar) to the sauce in small handfuls, stirring gently until each addition is melted before adding more; do not increase the heat to avoid a grainy sauce.
Taste and season the cheese sauce with kosher salt and pepper as needed.
Pour the cheese sauce over the drained pasta in the pot, add the blanched broccoli, and stir to combine.
Serve and, if desired, sprinkle the remaining grated cheddar over individual portions so it melts on contact for extra cheesiness.