Bring a large stockpot of water to a boil. Add pasta, and cook until al dente, stirring occasionally to keep the pasta from sticking, about 8 minutes. Transfer to a colander to drain, reserving 1/2 cup cooking liquid. Set pasta aside.
Heat 1 teaspoon olive oil in a large skillet over medium heat. Add garlic; saute until golden, stirring to avoid burning, about 2 minutes. Add broccoli and reserved cooking liquid; cook, stirring, until broccoli is tender and bright green, about 3 minutes. Add the salt, and season with pepper. Remove from heat.
Add reserved pasta to pan, and toss well to coat. Transfer to four serving bowls; drizzle each with 1/2 teaspoon olive oil, and garnish each serving with 1 1/2 teaspoons grated pecorino cheese.