Bring a large pot of salted water to a boil. Add the orecchiette and cook according to package instructions.

Meanwhile, in a large skillet, heat 1 tablespoon of the olive oil over medium high heat. Crumble the sausage into the skillet and cook, breaking apart with a spoon, until lightly browned, 5 to 6 minutes. Reduce the heat to medium and add the garlic; cook for 1 minute more.

Add the remaining 4 tablespoons of olive oil, chicken broth, broccoli, salt and red pepper flakes. Cook, stirring frequently and scraping the bottom of the pan to release the flavorful brown bits, until the broccoli is tender-crisp, 3 to 4 minutes. Stir in the butter until melted and simmer for a few minutes to reduce and concentrate the sauce.

Drain the pasta well and place back in the pasta pot. Add the sausage and broccoli mixture, along with half of the cheese, and toss to blend. Taste and adjust seasoning. Transfer to a serving platter or individual bowls and pass the remaining grated cheese at the table.

This dish can be frozen for up to 3 months and reheated on the stovetop. If the dish has dried out a bit in the freezer, add a little water or broth while reheating. Just take care only to reheat it and not continue to cook it.