In a pot of salted boiling water, cook potatoes until very tender. Drain. Mash with butter. Add milk and egg and combine until smooth. Season with salt and pepper. Add half of the cheese. Set aside.
With the rack in the middle position, preheat the oven to 180 °C (350 °F).
In a non-stick skillet, brown sausage meat in 30 ml (2 tablespoons) of butter, breaking meat with a wooden spoon. Place in a 28 x 20-cm (11 x 8-inch) baking dish. Set aside.
In the same skillet, brown onions and mushrooms in remaining (15 ml/1 tablespoon) of butter. Season with salt and pepper. Add cabbage and garlic and soften for 2 minutes. Deglaze with broth. Add peas. Spread over sausage meat and press lightly.
Spread mashed potatoes over vegetables. Sprinkle with remaining cheese. Bake for 45 minutes or until pie is golden brown. Let cool for 10 minutes before serving.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.