Heat the olive oil over medium heat in a large saucepan and brown sausage for 5 or 10 minutes to allow the meat to firm up. Remove to a utility platter until cool enough to handle.
To the same pan add onion, garlic and thyme leaves and saute until the onion becomes translucent. Stir in 8 cups water and 1 tablespoon balsamic vinegar. Add carrots and lentils, bring to a boil, reduce heat and simmer for about 30 minutes until the lentils begin to soften.
While the lentils are cooking, slice the sausage into 1/4-inch slices. Add them to the pot of lentils and simmer until the meat is cooked through, about 15 more minutes. Season soup with salt and pepper, to taste. Serve drizzled with extra-virgin olive oil.