For the Cotechino
Put the Cotechino in a large pot and cover it with water, then add the celery and a carrot (cut in rounds). While it is cooking, poke the cotechino with a toothpick, so that the excessive fat will come out. Cover the pot with a lid, and leave the Cotechino there for about 2 hours over medium heat. When the cotechino will be cooked, drain it and remove its skin. Then cut it into 1 cm-slices.
For the purée
Boil the potatoes for about 30 – 40 minutes. Peel them when still warm – you use a fork to help you. Then, mash them with a potato masher and place the mashed potatoes in a saucepan. Add butter and milk, and salt to taste. Over low heat, mix everything and let it cook (while stirring) for a few minutes, until you have a smooth and homogeneous purée.
For the lentils
Put the lentils in a large mixing bowl and cover them completely with water. Leave them soaking in the water for about 2 hours, changing the water a couple of times, to help them soften really well. After 2 hours, drain the lentils and set them aside. Chop the onion, the celery, the carrot and the shallot: sautée everything in a pan with oil, and stir carefully. Plate the lentils in the same pan and let them cook and flavor for about 40 minutes. When ready, turn off the heat.