Bring a large pot of salted water to a boil. Cook the noodles according to package instructions but reduce the cooking time by about 2 minutes. Drain and rinse under cool water to stop cooking.
Cut the broccoli into small florets. Mince the garlic and ginger.
In a bowl, whisk together garlic, ginger, 3/4 cup chicken broth, soy sauce, oyster sauce, honey, and cornstarch until smooth.
Heat olive oil in a large skillet over medium-high heat. Add ground beef, salt, and black pepper. Cook for 5–7 minutes, breaking up the meat until browned.
Add broccoli and the remaining 1/2 cup chicken broth. Stir, cover, and cook for about 3 minutes until the broccoli becomes bright green and slightly tender.
Stir the sauce again, then add it along with the cooked noodles to the skillet. Toss everything together and cook for 2–3 minutes until the sauce thickens and coats the noodles well. This step brings the classic flavor of beef and broccoli noodles together beautifully.
Garnish with scallions and sesame seeds before serving.