Melt 2 Tbsp butter in a bowl, then add the oyster sauce, soy sauce, brown sugar, and sesame oil. Stir to combine, then set the sauce aside.

Bring a large pot of water to a boil. Once boiling, add the spaghetti and cook until tender (about 7 minutes). Drain the spaghetti in a colander.

While the spaghetti cooks, begin the beef and broccoli. Add the remaining 1 Tbsp butter to a large skillet and melt over medium heat. Add the ground beef and cook until browned (5-7 minutes). Drain any excess fat if necessary.

While the beef cooks, mince the garlic. Add the minced garlic to the skillet with the browned beef and cook over medium heat for 1-2 minutes more, until the garlic has softened slightly.

Add the frozen broccoli florets to the skillet with the beef and garlic. Continue to sauté over medium heat just until the broccoli is no longer frozen. It will continue to cook in the next steps.

Finally, add the cooked and drained pasta and the prepared sauce to the skillet with the beef and broccoli. Cook and stir until everything is combined, coated in sauce, and heated through.

Slice the green onions and sprinkle over top just before serving, along with a pinch of crushed red pepper flakes, if desired.