Preheat the oven to 200°F (95°C).

Cook sausages in a skillet over medium-low heat until browned, about 5 minutes per side. Transfer to an oven-safe dish and keep warm in the preheated oven.

Place potatoes into a saucepan over medium heat, cover with water, and bring to a boil. Simmer until potatoes are tender, about 10 to 15 minutes. Drain and steam dry for a minute or two.

Mix in 1/4 cup butter, milk, dry mustard, salt, and black pepper into the potatoes. Mash until fluffy and smooth. Set aside.

Melt 1 tablespoon butter in a skillet over medium-high heat. Add onions and cook until translucent and slightly browned, about 8 minutes.

Pour beef broth and red wine into the skillet with onions. Bring to a boil and simmer until reduced to half its volume, about 10 minutes. Season with salt and pepper.