1. Warm oil in a medium skillet over medium-high heat. Add garlic and sauté until fragrant, about 30 seconds. Add sausage and cook, breaking up with a wooden spoon, until it loses its pink color, about 6 minutes. Add tomatoes with their juice and 1/4 tsp. salt and cook, stirring occasionally, until most of juice has evaporated and sauce has thickened, about 10 minutes.
2. While sauce is cooking, bring a large pot of salted water to boil. Stir in pasta and cook according to package label directions. In last 2 minutes of cooking time, stir in broccoli and continue to cook until pasta is al dente and broccoli is tender. Drain, return to pot and toss with sauce. Serve immediately, with shredded Parmesan on the side, if desired.