Preheat the oven to 325 degrees . Season the brisket with salt and pepper. In a large, heavy skillet, melt 1 teaspoon butter over medium-high heat. Working in batches, add the brisket and cook for 8 minutes; transfer to a large Dutch oven and reserve the skillet.
Add 2 teaspoons butter to the skillet. Add one-third of the onions and cook over medium heat until softened; push to the side of the pan. Repeat with the remaining onions, then cook over medium heat, stirring, until browned, about 8 minutes more. Transfer to the Dutch oven. Add 3 tablespoons butter to the skillet, over medium heat; add the flour and cook, stirring, until the mixture is golden. Whisk in the beer, then the beef broth; bring to a boil, whisking until smooth.
Pour the sauce over the meat and onions and bring to a simmer over medium heat. Cover the pot and cook in the oven for 2 hours. Remove the lid and cook until the meat is fork-tender, 30 minutes. Set aside and let rest for 30 minutes.
Meanwhile, in a large pot, place the potatoes in salted water to cover; bring to a boil, then simmer for 15 minutes. Drain and return to the pot. Add the remaining 4 tablespoons butter and mash. Beat in the milk and season with salt and pepper. Serve the stew over the mashed potatoes.