1. Preheat oven to 350 degrees with rack in upper third. Melt butter in a saucepan over medium-high heat. Whisk in flour; cook, whisking, 1 minute. Slowly whisk in broth and milk; bring just to a boil, whisking constantly. Remove from heat; season sauce with salt and pepper.

2. Heat oil in a skillet over medium-high heat; cook sausage until browned, about 2 minutes a side. Transfer to a 9-by-13-inch baking dish, scattering evenly.

3. Reduce heat to medium, add peppers to skillet, season with salt, and cook until browned and softened, about 7 minutes. Add vinegar; cook 30 seconds more, scraping up browned bits. Spoon evenly over sausage in dish.

4. In a bowl, combine potatoes, cheese, scallions, and sauce. Add to baking dish in an even layer. Cover with parchment-lined foil. Bake 40 minutes, then uncover and turn oven to broil; cook until top is well browned, about 8 minutes. Serve warm.