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, then drain and return to the pot.
While the pasta cooks, make the cheese sauce: melt the butter in a saucepan over medium heat.
Whisk the flour into the melted butter and cook, whisking, until the roux is golden and fragrant, about 2 minutes.
Combine the milk and heavy cream, then slowly add the mixture to the roux in small streams (about 1/4 cup at a time), whisking constantly until smooth and slightly thickened.
Stir in the cream cheese until fully melted and incorporated.
Remove the saucepan from the heat and gradually add about 12 ounces of the grated cheddar in small handfuls, stirring gently until completely melted to avoid a grainy texture.
Season the cheese sauce with kosher salt and pepper to taste.
Pour the cheese sauce over the drained pasta and add the blanched broccoli; stir gently to combine.
Serve immediately, stirring in or sprinkling the reserved grated cheddar as you plate for extra melty cheese.