In a covered 5- to 6-quart nonstick pan over high heat, bring 2 1/2 to 3 quarts water to a boil.
Meanwhile, rinse broccoli. Trim stems and cut crosswise into 1/4-inch-thick slices, then cut florets lengthwise into 1-inch pieces. Peel onion and cut into thin slivers. In a small bowl, mix broth, black bean sauce, and cornstarch.
Add spaghetti to boiling water and cook, uncovered, until barely tender to bite, 6 to 9 minutes. Pour into a colander, rinse with hot water, and drain.
Return pan to high heat and pour in oil. Add beef and stir often, breaking up with spoon, until lightly browned and crumbly, 1 to 2 minutes. Add onion and stir until it begins to brown, about 1 minute.
Add broccoli and 1/2 cup water; cover and stir occasionally until bright green, about 2 minutes. Stir broth mixture, add to pan, and stir until boiling.
Add noodles to pan; stir over medium heat until hot and coated with sauce, about 1 minute. Add soy sauce to taste. Pour into bowls.