Heat oil in a large cast-iron Dutch oven over medium-high. Add beef, and sprinkle with 1 teaspoon of the salt and 1/2 teaspoon of the pepper. Cook, stirring occasionally, until browned on all sides, about 6 minutes.
Add mushrooms, carrots, and onions, and cook over medium-high, stirring often, until vegetables begin to soften, about 5 minutes. Add flour and garlic, and cook, stirring constantly, 1 minute. Add beer, and simmer until reduced by about half, about 2 minutes. Add stock and remaining 1 teaspoon salt and 1/2 teaspoon pepper, and bring to a boil. Reduce heat to medium-low; cover and simmer until beef is very tender, about 1 1/2 hours.
Rub both sides of sourdough bread slices with cut sides of garlic clove. Melt 2 tablespoons butter in a large cast-iron skillet over medium-high. Once butter begins to foam, add 3 bread slices, turning immediately to ensure both sides are coated with melted butter. Cook until bottom is golden, about 1 minute. Turn bread slice, and cook until bottom is golden, about 1 minute. Repeat with remaining 2 tablespoons butter and 3 bread slices. Serve Buttery Garlic Bread immediately with Beef Stew.