Heat the oven to 425°F. Line a rimmed baking sheet with parchment and lightly coat it with oil or cooking spray.
For the meatballs, stir the breadcrumbs and milk in a large bowl to form a loose paste. Add the ground beef, sausage, Parmesan, egg, salt, pepper, parsley, and 1 small clove of grated garlic. Mix gently with your hands just until combined. Scoop into 1-inch portions (about 1 tablespoon each), roll into balls, and arrange on the sheet. Tip: Lightly oil your hands or a small scoop for perfectly round meatballs.
Chill the shaped meatballs in the refrigerator for 10 minutes to help them hold their shape, then bake until lightly browned and almost cooked through, 12 to 14 minutes, rotating the pan once halfway through.
While the meatballs bake, start the soup. Warm the olive oil in a large Dutch oven over medium heat. Add the onion, celery, and carrots with a pinch of salt. Cook, stirring occasionally, until the vegetables are glossy and beginning to soften, 8 to 10 minutes.
Stir in the 3 minced garlic cloves and the dried oregano; cook until fragrant, about 30 seconds. Pour in the chicken broth and bring to a gentle boil, then lower to a lively simmer for 5 minutes.
Add the pasta to the simmering broth and cook uncovered, stirring occasionally, until just shy of al dente, 7 to 9 minutes depending on shape.
Slide the baked meatballs and any juices into the pot. Reduce the heat to low and simmer 5 minutes to marry the flavors. Stir in the spinach until just wilted, 1 to 2 minutes. Taste and season with salt and more pepper as needed.
Ladle into warm bowls and finish with a generous shower of Parmesan. Serve immediately.