1. Place potatoes in a large pot; cover with water. Add 1 Tbsp. salt. Bring to a boil and cook until tender, 20 to 25 minutes. Drain, return potatoes to pot, and mash with butter. Mix in milk and mustard. Season with salt and pepper. Cover to keep warm.
2. Prick sausages all over with a fork. Warm oil in a skillet over medium-high heat; brown sausages on both sides, about 5 minutes. Remove to a plate.
3. Add onion to skillet and sauté until softened, about 3 minutes. Add flour and sauté 1 minute. Slowly stir in broth and cook, whisking, until sauce has thickened, about 3 minutes; season with 1/2 tsp. salt. Return sausages to skillet, cover, reduce heat to low and simmer until sausages are warmed through, about 5 minutes. Slice and serve over mashed potatoes, with extra mustard for dipping.