In a large dutch oven, heat the EVOO over medium-high heat. Add the sausage and cook until browned; transfer to a bowl with a slotted spoon.
Add the onion and carrot to the pot and cook until slightly softened, 3 to 4 minutes. Stir in the garlic and rosemary and cook until fragrant, about 1 minute. Add the lentils, broth and 3 cups water. Bring to a boil, lower the heat and simmer until the lentils are tender, 25 to 30 minutes.
Stir in the escarole and sausage, cover the pot and cook until the escarole is wilted and the sausage is heated through, 3 to 4 minutes. Season with salt and pepper.