In a covered 5- to 6-quart pan over high heat, bring 2 1/2 to 3 quarts water to a boil.
In a bowl, mix beef, 1/2 cup onions, egg white, bread crumbs, 1/2 tablespoon garlic, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Shape mixture into 12 equal balls (about 1 3/4 in. each).
Place meatballs in a single layer, without crowding, in an 11- to 12-inch nonstick frying pan over high heat; turn as needed to brown on all sides, about 5 minutes total per batch. As meatballs are browned, transfer to a plate with a slotted spoon.
Add remaining 1 1/2 cups onions and 1/2 tablespoon garlic to frying pan; stir often until onions begin to brown, about 5 minutes. Add tomato purée, wine, broth, basil, and sugar; stir until boiling.
Add browned meatballs to sauce; when sauce boils, cover, reduce heat, and simmer, stirring occasionally, until meatballs are no longer pink in the center (cut to test), 8 to 10 minutes.
Meanwhile, add spaghetti to boiling water and cook until barely tender to bite, 7 to 9 minutes; drain well.
Mound spaghetti in a wide bowl. Pour meatballs and sauce over pasta. Sprinkle with 1/4 cup cheese and parsley; serve with salt, pepper, and additional cheese to add to taste.