Preheat the oven to 450 degrees F. Brush a 9-by-13-inch baking dish with oil.
Bring a large pot of salted water to a boil. Cook the spaghetti according to the package instructions for al dente. Strain, add back to the pot and toss with some oil.
Meanwhile, make the meatballs: Soak the bread with 2 tablespoons of the milk in a medium bowl, tossing occasionally, until completely soft, about 5 minutes. While the bread softens, combine the beef, pork, 1 of the eggs, 1/3 cup of the Parmesan, the minced garlic, 1 teaspoon salt and a few grinds of pepper in another medium bowl. Add the soaked bread to the meat mixture. Gently incorporate all the ingredients with your hands until well combined.
Brush a baking sheet with oil, and drop teaspoonfuls of the meatball mixture onto it. Brush your hands with a little oil, roll the mounds into balls and space them out evenly on the baking sheet. Bake until golden brown and almost completely cooked through, about 10 minutes.
Lower the oven temperature to 350 degrees F. Add the meatballs, mozzarella and sun-dried tomatoes to the cooked pasta, toss to combine and transfer to the prepared baking dish.
Whisk together the marinara sauce, the remaining 5 eggs, 2/3 cup Parmesan and 1/4 cup milk, the sliced garlic and 1 teaspoon salt in one of the empty bowls. Pour the sauce mixture over the spaghetti mixture, and press the spaghetti into a flat, even layer. Sprinkle the parsley over the top. Cover with foil, and bake until the casserole is hot and almost completely set, about 30 minutes. Remove the foil, and continue baking until the top of the casserole is set, about 10 minutes more. Let cool for a few minutes. Cut into squares, and serve warm.
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