Bring a medium saucepan of salted water to a boil.
In a large Dutch oven or heavy pot, heat the oil over medium-high. Add the carrots, broccoli stems and garlic, season with salt and pepper, and cook, stirring occasionally to keep the garlic from scorching, until softened, 3 to 5 minutes.
Add the chicken broth and bring to a boil, then reduce heat and simmer until the broth is flavorful, 5 to 7 minutes. Taste and season with salt and pepper as it simmers.
Once the broth begins to simmer, add the pasta to the boiling water and cook until al dente. Drain the pasta and divide it among bowls.
While the pasta cooks and the broth simmers, add little meatballs and plop them into the broth. Return to a boil, add the broccoli florets, then reduce heat and simmer, stirring gently and occasionally, until the meatballs are heated through and the florets are bright green, 5 to 7 minutes. Season to taste with salt and pepper. Spoon the soup over the pasta. Finish with Parmesan.