Cook the 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 preheated oven to keep warm.
Place potatoes into a saucepan over medium heat, cover with water, and boil gently until potatoes are tender, 10 to 15 minutes. Drain and allow to steam dry for a minute or two. Mix in 1/4 cup of 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 the onions until translucent and just starting to brown, about 8 minutes. Pour in the beef broth and red wine; boil the mixture down to about half its volume, about 10 minutes. Season with salt and black pepper. To serve, place a sausage onto a serving plate with about 1/2 cup of mashed potatoes. Pour the onion gravy over the sausage and potatoes.