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Cook onions in extra-virgin olive oil in a 12-inch heavy skillet over medium heat, stirring occasionally, until softened, about 10 minutes. Add garlic and cook, stirring occasionally, until softened, about 3 minutes. Transfer to a large bowl to cool.
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Soak bread in milk in another bowl until soft, about 5 minutes. Firmly squeeze bread to remove excess milk, discarding milk.
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Stir together cooled onion mixture, bread, eggs, parmesan, parsley, oregano, lemon zest, 5 1/2 tsp salt, and 1 1/2 tsp pepper until combined. Add meats to bread mixture, gently mixing with your hands until just combined (do not overmix).
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Form meat mixture into about 70 (1 1/2-inch) balls with dampened hands, arranging meatballs on 2 large baking sheets or in shallow baking pans.
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Heat olive or vegetable oil (1 cup) in a 12-inch heavy skillet (preferably nonstick) over medium-high heat until hot but not smoking, then brown meatballs in 4 or 5 batches (without crowding), turning frequently, about 5 minutes per batch. Return to baking sheets.
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Add meatballs to sauce and gently simmer, covered, stirring occasionally, until cooked through, 20 to 30 minutes. (If pot is not large enough, divide meatballs and sauce between 2 pots.)