1/4 cup Parmigiano-Reggiano grated, plus more for serving
2 tablespoons unsalted butter
Remove the meat from the casings and crumble it into a small bowl, discarding the casings.
Heat a large nonstick skillet over medium high heat and add the oil. When it’s hot, add the sausage and break it up into small pieces using a wooden spoon. Cook, stirring, until no longer pink— about 3 minutes. Add the garlic and red pepper and stir, cooking 1 minute more, until fragrant.
Stir in broccoli and cook for about 2 minutes, until it begins to wilt. Pour in chicken stock and allow it to come to a boil. Lower the heat to medium low and simmer, covered, until the broccoli is tender and bright green—about 3 minutes.
Uncover the pan and simmer a few minutes more, until the sauce has reduced and thickened up a bit. While the stock is reducing, bring a medium pot of water to a gentle boil. Add a generous pinch of salt. Cook orecchiette 3-4 minutes until al dente and drain. It will cook in about the same time, whether you choose to cook it from fresh or frozen.
Add cooked pasta, Parmigiano and butter to the pan and toss to well coat the pasta in the sauce. Continue to toss and cook for a few mins more, until everything is nicely coated and glossy. Season with salt if needed—I find it doesn’t need any additional (the sausage and cheese are already salty). Serve with extra cheese sprinkled on top. Eat right away.