Prepare the vegetables: Peel and chop carrots, onion, and celery.
Heat oil in a large pot over medium heat until shimmering.
Add carrots, onion, and celery and stir to coat. Cook over medium heat, stirring occasionally, until veggies are slightly softened.
Stir in garlic and cook for an additional minute.
Stir in chicken broth, scraping up any browned bits from the bottom of the pot.
Bring to a boil, then lower the heat and simmer until the veggies are softened, about 20 minutes.
Meanwhile, preheat oven to 425 F.
Line a baking sheet with parchment paper.
Prepare the meatballs: In a bowl, mix to combine ground beef, sausage, Parmesan, parsley, breadcrumbs, egg, garlic, salt, and pepper using gloved hands.
Use a 1-inch scoop to portion meatballs onto the prepared baking sheet, spacing apart.
Roll each scoop into a smooth ball.
Bake for 8 to 10 minutes, until meatballs are browned and cooked through.
When the soup is ready, add pasta and cook until just al dente, about 8 minutes.
Slowly stir in spinach and cook until wilted.
Stir in hot meatballs.
Serve the soup, garnished with shaved Parmesan if desired.