In a Dutch oven (or other 5-quart pot with a tight-fitting lid), cook bacon over medium-low heat until browned and crisp, 8 to 10 minutes. Pour off all but 1 tablespoon fat.
Add onion and carrots; cook until softened, about 5 minutes. Stir in garlic, and cook until fragrant, about 30 seconds. Stir in tomato paste, and cook 1 minute.
Add lentils, thyme, broth, and 2 cups water. Bring to a boil; reduce to a simmer. Cover; cook until lentils are tender, 30 to 45 minutes.
Stir in vinegar, 1 1/2 teaspoons salt, and 1/4 teaspoon pepper. Serve immediately.
Note: If you prefer to leave out the bacon, skip step 1, and cook the onions and carrots in 1 tablespoon olive oil.