Peel and chop onion, carrots, and garlic. In a 4- to 5-quart nonstick pan over high heat, stir vegetables in olive oil until onion is limp, 4 to 5 minutes.
Drain and rinse garbanzos. Add garbanzos and broth to pan. Cover and bring to a boil; reduce heat to low and simmer until carrots are tender when pierced, 5 to 7 minutes.
Meanwhile, remove and discard any yellow or wilted leaves from watercress; rinse and drain watercress. Pluck off about 2 cups tender sprigs (about 3 in. long). Chop enough remaining watercress, including stems, to make about 1/2 cup; reserve remainder for another use or discard.
Working in batches, whirl all but 4 1/2 cups garbanzo mixture in a blender until smooth, holding lid down with a towel. Return purée to pan with remaining garbanzo mixture.
Stir often over high heat until boiling; add watercress sprigs and chopped watercress and stir until wilted, about 30 seconds. Add salt and pepper to taste. Ladle soup into bowls.