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. 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.
Once you've poached the chicken in the broth it doubles in flavor. Freeze or refrigerate any leftovers for other recipes.