In a large pot or Dutch oven, heat the olive oil over medium-high heat. Add the carrots and onion and cook, stirring occasionally, until softened and browned, about 8 minutes. Add the mushrooms and cook, stirring occasionally, for 2 minutes. Transfer the vegetables to a medium bowl. Add the beef to the pot and cook over medium-high heat, breaking it up, until browned, about 3 minutes; season with salt and pepper. Add the vegetable mixture, chicken broth and tomatoes.
Stir in the pasta and bring to a boil, stirring occasionally. Lower the heat to medium-low, cover and simmer until the pasta is tender, 10 to 12 minutes; season with salt and pepper. Serve topped with the sour cream.