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.
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. Rinse with cold water. Return the water to a boil, add the fusilli and cook, stirring occasionally until al dente to tender but not mushy, about 8 minutes.
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.
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.
