Pierce the cotechino with a fork in several places & let it soak in water in a large pot for 1 hour.
Place chopped onion, bay leaf, peppercorn, thyme, and star anise in a large pot with cotechino. Cover with water & bring all to a boil. Reduce heat to low & let simmer for one hour.
Remove sausage from water & set aside. Reserve the water for the lentils.
Heat oil in a medium saucepan over medium heat. Add onion & carrots. Reduce heat to medium-low & cook, stirring occasionally until softened, about 5 minutes. Add celery, garlic, & cook, stirring 2-3 minutes more.
Add lentils & bay leaf, stir & combine. Add reserved water from the cotechino to cover lentils by 1 inch. Bring to a gentle simmer & cook, stirring occasionally & adding more liquid to keep the lentils covered until tender, 30-40 minutes. Drain liquid from pot; remove & discard garlic & bay leaf; season to taste with salt & pepper.
Spoon the lentils onto serving plates.
Transfer sausage to cutting board & cut into 1/3 inch thick slices; arrange on top of lentils. Drizzle with balsamic glaze or make creamy mustard sauce.