Preheat oven to 350°. Place breadcrumbs in a small bowl. Drizzle the milk over and let stand until absorbed.
Heat 2 tablespoons of oil in a large skillet over medium-high heat. Add the onion and garlic; cook until translucent, about 3 minutes. Set aside.
Place the ground meats in a large bowl. Add breadcrumb mixture to meats along with the reserved onion and garlic, whole eggs, yolks, parsley, oregano, and cheese. Season with salt and pepper. Using your hands, mix until just combined. Moisten hands with water, and roll meat mixture into 3-inch meatballs. Place on a baking sheet.
Heat 1/4 cup of oil in a large skillet over medium heat. Working in batches, add meatballs, and brown on all sides, about 5 minutes. Finish cooking in sauce, or transfer to a baking sheet fitted with a rack to drain. Bake until cooked through, 12 to 15 minutes.