Discard sausage casings. In a 10- to 12-inch nonstick frying pan over high heat, stir sausages until broken up and no longer pink, about 5 minutes.
Add broccoli florets; stir until sausages are well browned and broccoli is tender-crisp to bite, 3 to 5 minutes more. Pour into a bowl; keep warm.
Add potatoes and onion to pan, reduce heat to medium-high, and stir until onion is limp, 2 to 3 minutes.
Add sage and milk to pan, and stir just until potatoes are tender to bite, 8 to 10 minutes more.
Pour potato mixture onto a platter; spoon sausage and broccoli mixture over it. Add salt to taste.