Cook pasta in a large pot of salted boiling water 2 minutes less than directed on package. Drain.
Meanwhile, cook garlic in oil in a large lidded pot over medium heat until golden, 3 to 4 minutes. Add broccoli and cauliflower and cook 1 minute. Stir in 1/4 teaspoon salt, and cook, covered, 3 minutes. Add broth, butter, beans, pepper, and remaining 1/4 teaspoon salt, and cook until beans are warm and sauce thickens slightly, about 5 minutes.
Add pasta and cook, stirring, until heated through, about 1 minute. Sprinkle with cheese.