Brown beef in a 6-quart Dutch oven over medium-high heat for 7 minutes, breaking it into small crumbles and cooking it to 160°F. Drain excess fat.
Stir in garlic and ginger for 30 seconds. Add broth, soy sauce and brown sugar and bring to a boil.
Stir in broken spaghetti and simmer uncovered for 7 minutes, stirring often to separate the strands. Add reserved water in small amounts if the bottom becomes dry.
Fold in broccoli, cover and cook over medium-low heat for 4 to 5 minutes until noodles are tender and broccoli is bright green and just tender.
Mix cornstarch with the 1 tablespoon cold slurry water. Stir into the pot with rice vinegar and simmer uncovered for 1 minute until the sauce lightly coats the noodles.
Remove from the heat and fold in sesame oil. Divide among 4 bowls and serve immediately.