Brown the ground beef in a large pot or Dutch oven over medium-high heat. Drain off any excess grease.
Add the minced garlic and grated ginger to the pot with the beef. Cook for 1 minute until fragrant.
In a small bowl, whisk together the beef broth, soy sauce, brown sugar, oyster sauce, and cornstarch until the cornstarch is dissolved.
Pour the sauce mixture into the pot. Bring the mixture to a simmer, stirring occasionally.
Add the ramen noodles (breaking them slightly if needed) and the broccoli florets to the simmering liquid.
Cover the pot, reduce the heat to medium-low, and cook for 5 to 7 minutes, or until the noodles are tender and the broccoli is crisp-tender. Stir halfway through cooking to separate the noodles.
Stir in the sesame oil. If the sauce seems too thick, add the 1/4 cup of water to reach your desired consistency.
Serve immediately directly from the pot.
Notes
You can substitute ground turkey or thinly sliced flank steak for the ground beef.
For a spicier dish, add 1 teaspoon of sriracha or red pepper flakes with the sauce ingredients.
If you prefer softer broccoli, add it during the last 2 minutes of cooking.