Pulse the sandwich bread in a food processor to form coarse crumbs. Add the beef, 3 tablespoons onion, mushroom, 1 tablespoon garlic, parsley, and egg white; season with salt, pepper and nutmeg; pulse again to finely chop and combine. Portion into about 20 meatballs (about 1-inch in diameter) and place on a plate.
In a medium saucepan, heat the oil over medium heat; add the remaining garlic and onion. Cook, stirring occasionally, until tender, about 10 to 12 minutes. Stir in the tomatoes and broth. Tie sprigs of basil and thyme together with kitchen twine; add to the tomatoes. Bring to a simmer and cook, stirring occasionally, 20 minutes. Add the meatballs and continue to simmer, without stirring, until the meatballs are partially cooked, about 10 minutes. Turn the meatballs; cook 10 minutes more or until fully cooked.
Meanwhile, cook the pasta according to package directions in boiling salted water; serve the meatballs and sauce over the spaghetti.