Heat 1 to 2 Tbsp. olive oil in a large skillet over medium heat. Add chicken sausage and cook until browned on both sides, about 6 to 8 minutes. Transfer to a bowl.Add remaining 2 Tbsp. olive oil, along with fennel, onion, and carrots. Cook until softened and nicely caramelized, about 8 minutes. Stir in tomato paste and garlic, stir to coat vegetables. Cook until the tomato paste turns brick-red and the garlic is aromatic, about 2 to 3 minutes.
Add lentils, broth, rosemary, salt, and pepper; bring mixture to a boil. Add sausage back to pan and reduce heat to maintain a gentle simmer. Cover and cook until the lentils are al dente, about 45 to 50 minutes. Remove lid and stir in kale. Cook, uncovered, until the greens are wilted, about 2 more minutes. Stir in grated cheese.
Taste and adjust seasonings as needed. Spoon into bowls and top with extra grated cheese and a drizzle of extra-virgin olive oil.
