Heat oil in a large saucepan over medium-high heat, swirling to coat pan. Add chopped onion and salt and sauté, stirring occasionally, until golden brown, about 20 minutes. If onions start to cook too quickly, add about ¼ cup water to slow the cooking and allow carmalization. Add carrots and dried thyme, and continue to cook for 5-7 more minutes. Add tomatoes, beans, a few grinds of fresh pepper, broth, and rosemary springs to pan. Bring to a boil, lower heat to simmer, and cook 5 minutes. Add broken spaghetti pieces, and cook for about 10 minutes until pasta is al dente (cooking time may vary based on pasta shape). Remove and discard rosemary stems. Taste for seasoning, adding more salt or pepper as desired. Add cooked sausages to the soup, or to individual bowls for serving. Garnish with additional rosemary if desired.