Start the pasta cooking according to package directions in a separate pot of salted boiling water—this allows it to cook while you prepare the other components. While the pasta cooks, mince the garlic cloves finely and grate the fresh ginger. Cut the broccoli into bite-sized florets. Having everything prepped and ready before you start cooking the beef will keep the cooking process smooth and prevent the beef from overcooking while you chop.
In a small bowl, whisk together the soy sauce, brown sugar, rice vinegar, 1/2 cup water, cornstarch, and sesame oil. Stir until the cornstarch is fully dissolved and the brown sugar is incorporated. This creates a smooth, thickening sauce that will coat the beef and broccoli beautifully. Setting this aside now means you can pour it in quickly during the cooking phase without any fumbling.
Heat the oil in a large skillet over medium-high heat until shimmering. Add the ground beef and cook, breaking it apart with a spoon as it browns, for about 3-4 minutes until most of the pink is gone. Once the beef is mostly cooked through, add the minced garlic and grated ginger, stirring constantly for about 30 seconds to bloom their flavors—this quick cooking releases their aromatic oils and prevents them from burning.
Add the broccoli florets from Step 1 to the skillet with the cooked beef. Pour in 1/4 cup water, cover the skillet, and reduce heat to medium. Steam for 2-3 minutes until the broccoli is bright green and tender-crisp but still has a slight bite to it. I like to peek after 2 minutes because broccoli can go from perfect to mushy quickly, and you want it to retain some texture.
Uncover the skillet and pour in the prepared sauce from Step 2. Stir everything together and cook uncovered for 2-3 minutes, stirring occasionally, until the sauce thickens and coats the beef and broccoli. By this time, your pasta from Step 1 should be cooked and drained. Add the cooked pasta to the skillet and toss everything together until well combined and evenly coated with sauce. Serve immediately while everything is hot and the sauce is glossy.