Serve four food groups in one delicious dish: corkscrew pasta, broccoli, chicken, and a sprinkling of Parmesan.
Bring broth to a boil in a medium saucepan. Add chicken. Reduce heat to medium-low; cover. Simmer until cooked through, about 10 minutes. Remove chicken; let cool. Using a fork, shred meat into bite-size pieces. Toss with 1 teaspoon oil and 1/4 teaspoon salt. Set aside.
Bring a large pot of water to a boil; add salt. Add broccoli, and cook until bright green and tender, about 3 minutes. Remove with a slotted spoon; drain, and pat dry with paper towels. Add pasta to boiling water; cook until al dente. Drain.
Heat remaining 1/4 cup oil in a large skillet over medium heat until hot but not smoking. Add cooked broccoli and the garlic. Cook, stirring often, 2 minutes. Stir in shredded chicken and cooked pasta. Cook 1 minute more. Remove garlic cloves. Transfer pasta mixture to a large platter. Serve with cheese.