Combine the beef, pork, eggs, breadcrumbs, dried oregano, grated garlic, Pecorino Romano, 1/2 teaspoon of the salt and pepper to taste in a large bowl. Using your hands, thoroughly mix the ingredients together until evenly incorporated. Roll into meatballs slightly larger than golf balls, setting them on baking sheets as you work.
Heat the oil over medium heat in a large Dutch oven until it shimmers. Working in batches and taking care not to crowd the pan, saute the meatballs, turning frequently, until browned on all sides, about 5 minutes. Using a slotted spoon, transfer from the pan back to the baking sheet.
Reduce the heat to medium-low, add the onion to the pan, and saute, stirring occasionally, until fragrant, about 2 minutes. Add the sliced garlic, stir, and cook 1 minute more. Add the wine, increase the heat, and bring to a boil, then reduce the heat and simmer for 2 minutes. Add the tomatoes, 4 of the basil leaves, the remaining 3/4 teaspoon salt and the meatballs along with any accumulated juices. Simmer, partially covered, until the flavors come together, about 1 hour 20 minutes. Taste and adjust the seasonings.
Meanwhile, cook the spaghetti in two batches in boiling salted water according to package directions. Drain and serve with the meatballs and sauce ladled on top. Garnish with torn basil and freshly grated Pecorino-Romano.