Bring a large pot of salted water to a rolling boil. Add the orecchiette and cook according to the package instructions until al dente. Drain well and set aside.
While the pasta is cooking, heat 1 tablespoon of the olive oil in a large skillet over medium‑high heat. Crumble the sausage into the skillet and cook, breaking it apart until it’s lightly browned (around 5‑6 minutes). Then add the minced garlic and cook for another minute.
Into the same skillet, add the remaining 4 tablespoons of olive oil, chicken broth, broccoli florets, salt and red pepper flakes. Cook, stirring frequently, scraping up any browned bits from the pan bottom, until the broccoli is tender‑crisp (about 3‑4 minutes). Then stir in the butter and allow the sauce to simmer for a few minutes to thicken slightly.
Return the drained pasta to its pot and pour in the sausage‑broccoli mixture along with half the grated pecorino. Toss everything well to combine, taste and adjust seasoning if needed. Transfer to a serving bowl or individual bowls, and top with the remaining cheese. Serve immediately.