Bring 6 cups of water to a boil and add the salt. Add the lentils, carrots, garlic, and sage and boil until the lentils are tender yet firm, about 25 minutes. Drain and place in a mixing bowl. Add the oil and vinegar, season with salt and pepper, and mix well. Set aside.
Prick the sausages with a pin several times. Place in a large pot of cold water and bring to a boil over medium heat. Reduce the heat to a very low boil and cover the pot. Cook for 1 1/2 hours.
Place the marinated lentils on a large serving platter to form a bed for the cotecchino. Drain the sausages and cut into 1/4-inch rounds. Arrange the sausage on the lentils and serve hot or at room temperature.
Recipe courtesy of Holiday Food, Mario Batali, Clarkson Potter. Photographs by Quentin Bacon.