Broccoli rabe: Roughly chop broccoli rabe, including the florets, leaves, and stems. Blanch in a large pot of salted boiling water, 2–3 minutes, then transfer with a slotted spoon to an ice bath for 2–3 minutes to stop the cooking. (Don't drain the water; keep it boiling to cook the pasta.) Remove broccoli rabe from ice bath and squeeze out excess water with your hands. Set aside.
Sausage: Set a medium skillet over high heat and add olive oil. When the oil is warm, add the sausage, breaking it up with a wooden spoon. Cook until brown, 5 minutes. Meanwhile, thinly slice the garlic and add to the sausage. Stir and cook 1 more minute. Mix the broccoli rabe into the sausage and garlic. Add salt and pepper to taste, followed by chilies, and continue cooking 3–4 more minutes.
Pasta: Put the orecchiette in the boiling water and cook about 2 minutes less than what the box advises (it will be al dente, says chef Mantuano, though not as chewy as other pasta shapes). Stir occasionally so the pasta doesn't clump.
Sauce: Add chicken stock to the sausage mixture to deglaze the pan, scraping up the brown bits on the bottom. Turn off heat and set aside.
Assembly: Reserve ½ cup pasta water, drain orecchiette, then return to the pot. Add sauce to the pot and stir. Cook over low heat, gently tossing the pasta with the sauce for 1–2 minutes to allow it to "marry"—or absorb—the sauce. (Add pasta water bit by bit if the sauce seems too dry.) Taste for seasoning and adjust if necessary. Transfer to a platter or bowl. Garnish with Pecorino Romano and serve immediately, with more cheese at the table.