Cut the sausage in half lengthwise. Slice into ½-inch-thick half-moon pieces. Add the sausage to a 6-quart stockpot over medium heat. Cook the sausage until it is browned, 5-8 minutes.
Add the butter, the onion, celery, and carrot. Stir and sauté until the onion is soft but not brown, 4-6 minutes. Add the garlic, salt, black pepper, Italian seasoning, and red pepper flakes. Stir until it is fragrant, 1 minute.
Add the tomato paste. Stir the paste into the vegetables and cook until it is darkened in color and beginning to coat the bottom of the pan, 3-4 minutes.
Add the lentils, rosemary, diced tomatoes, and chicken stock. Stir, bring to a boil, then reduce to a simmer and cover. Cook until the lentils are tender, 25-35 minutes.
Add the balsamic vinegar and taste for seasoning. Serve immediately.
