In a 5- to 6-quart pan over high heat, frequently stir mushrooms and 1/4 cup broth until liquid evaporates and mushrooms are lightly browned, about 5 minutes.
Meanwhile, in a bowl, beat egg to blend. Add beef, veal, bread crumbs, onion, and 1/4 cup broth. Mix well.
To mushrooms, add remaining broth, 2 cups water, tomatoes (with juice), tomato sauce, and Italian seasoning. Drop rounded tablespoons of meat mixture into pan. Bring to a simmer, turn meatballs over gently, and cook until no longer pink in center of thickest part (cut to test), 5 to 7 minutes. With a slotted spoon, put meatballs in a bowl.
Turn heat to high and add spaghetti to pan. Boil, stirring often, until pasta is tender to bite, about 10 minutes.
Return meatballs and any juice to pan. Stir gently just until hot, about 2 minutes.
Spoon spaghetti and meatballs into wide bowls or onto plates. Sprinkle with basil and cheese.