Ingredients

• 2 boneless, skinless chicken breast halves (each about 8 ounces)
• 3 cups chicken broth, homemade or low-sodium canned
• 1 tablespoon kosher salt
• 6 cloves garlic, 2 smashed and 4 minced
• 2 sprigs flat-leaf parsley
• 3 slightly heaping cups small broccoli florets (about 7 ounces)
• 12 ounces spinach fusilli pasta (about 3/4 of a pound box)
• 3 tablespoons extra-virgin olive oil
• 1/4 teaspoon chile flakes
• 1 cup cherry or grape tomatoes, halved
• 2 tablespoons unsalted butter, diced
• 1/2 cup grated Pecorino Romano, plus more to pass
• Freshly ground black pepper

Instructions

1. In a small saucepan with a tight-fitting lid, combine the chicken, broth, 1 teaspoon of the salt, smashed garlic, and parsley. Bring just to a boil, remove from the heat, and set aside, covered, until just cooked through, about 15 minutes. Remove the chicken, strain the broth. Pull the chicken into strips and put in a bowl with about 1/4 cup of the broth. Reserve the rest of the chicken broth.
2. Bring a large pot of cold water to a boil over high heat, then salt it generously. Add the broccoli and cook until crisp-tender, about 2 minutes. Transfer broccoli with a slotted spoon to a bowl. Return the water to a boil, add the fusilli and cook, stirring occasionally until al dente to tender but not mushy, about 8 minutes.
3. While the pasta cooks, combine the minced garlic, oil, and chile flakes in a large skillet. Cook over medium-high heat, stirring, until lightly browned and fragrant, about 2 minutes. Add the tomatoes and cook, stirring, until somewhat soft, about 1 minute. Add 1/2 cup of the broth, the remaining 2 teaspoons salt, and stir in the butter and 1/4 cup of the pecorino until incorporated. Stir in the chicken and broccoli, bring to a simmer, and remove from the heat.
4. Drain the pasta in a colander in the sink. Transfer to a bowl and toss with the remaining 1/4 cup pecorino. Pour the sauce over the pasta, season with pepper, and toss to combine. (If the sauce around the pasta seems "tight", add a little more broth, until it loosely naps the fusilli.) Serve in large bowls and pass additional cheese at the table, if desired.