Cook sausage links in a skillet over medium-low heat until browned on all sides, about 5 minutes per side. Transfer to an oven-safe dish and move to the oven to keep warm.
Cover potatoes with water in a saucepan and boil gently until tender, 10–15 minutes. Drain and allow to steam dry for a minute or two. Mix in ¼ cup butter, milk, dry mustard, salt, and black pepper; mash until fluffy and smooth. Set aside.
Melt 1 tablespoon butter in a skillet over medium-high heat. Cook onions until translucent and starting to brown, about 8 minutes. Pour in beef broth and red wine; boil down to about half its volume, about 10 minutes. Season with salt and black pepper.
To serve, place a sausage on a plate with about ½ cup mashed potatoes. Pour onion gravy over the sausage and potatoes.
