In a large bowl, combine the beef, bread crumbs, egg, 1 teaspoon salt and 1/4 teaspoon pepper. Mix well and form into 1 1/2-inch meatballs.
In a large skillet, heat the olive oil over medium-high heat. Working in 2 batches, add the meatballs and cook, shaking the skillet often, until lightly browned, 5 to 7 minutes. Transfer to paper towels to drain.
Pour off all but 1 tablespoon of the fat in the skillet and add the carrots, onion and celery. Cook, stirring constantly, until softened, about 3 minutes. Stir in the flour and paprika and cook for 1 minute. Pour in the beef broth and bring to a boil over high heat, scraping up any browned bits from the bottom of the pan; season to taste with salt and pepper. Add the reserved meatballs and caraway seeds, reduce the heat to medium-low and simmer for 5 minutes.
Serve with buttered noodles.