Preheat oven to 350°. Line an 8-in. square baking pan with parchment paper, letting ends extend up sides.
Whisk together the first five ingredients. Add apples and beer; mix just until blended (do not overmix; batter will be thick). Transfer to prepared pan. Bake until a toothpick inserted in center comes out clean, 40-45 minutes.
Lifting with parchment paper, immediately remove cake from pan. Brush all sides with melted butter. Cool. Meanwhile, combine all icing ingredients. Spread or drizzle over cooled cake. Yield: 12 servings.