Place potatoes and 2 tsp. salt in a large, deep skillet and cover with cold water by 1 inch. Bring to a boil over high heat, reduce to a simmer and cook until potatoes are just tender, about 8 minutes. Drain. Wipe out skillet.
Melt butter in same skillet over medium heat. Add sausage. Cook, stirring occasionally, until browned on both sides, about 3 minutes. Using a slotted spoon, remove to a bowl. Add vegetables to skillet; season with salt and pepper. Cook over medium heat, stirring occasionally, until mushrooms have browned and vegetables are tender, about 8 minutes.
Whisk eggs with milk in a medium bowl; season with salt and pepper. Add sausage and potatoes to vegetables in skillet and stir to combine. Pour in egg mixture and cook, stirring constantly, until eggs are set but still creamy, about 1 minute. Serve immediately.
Keep warm. If the eggs are done before you're ready to eat, pop the (ovenproof) skillet in the oven on low heat to buy some time.