Allow butter and eggs to stand at room temperature for 30 minutes.
Meanwhile, line twenty four 2-½-inch muffin cups with paper liners.
In a medium bowl stir together flour, baking powder, and salt. Set aside.
Preheat oven to 350 degrees.
In a large mixing bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Gradually add sugar, about ¼ cup at a time, beating on medium speed until combined.
Scrape sides of bowl; beat on medium speed about 2 minutes more or until light and fluffy.
Add eggs, one at a time, beating well after each addition.
Beat in root beer extract and vanilla.
Alternately add flour mixture and root beer to butter mixture, beating on low speed after each addition just until combined.
Spoon batter into prepared muffin cups, filling each two-thirds to three-fourths full.
Bake for 15 to 17 minutes or until a toothpick inserted in centers comes out clean.
Cool cupcakes in muffin cups on wire racks for 10 minutes.
Remove cupcakes from muffin cups.
Cool completely on wire racks
Frosting
Allow butter to stand at room temperature for 30 minutes.
Allow ice cream to stand at room temperature for 10 minutes.
In a large mixing bowl beat butter with an electric mixer on medium speed until fluffy.
Beat in ½ cup of the ice cream and the vanilla.
Gradually beat in powdered sugar. If necessary, stir in enough of the remaining ice cream, 1 tablespoon at a time, to make frosting piping or spreading consistency.
Notes
The frosting recipe may need to be doubled depending on if you pipe or spread the frosting on. Spreading the frosting on doesn't use as much.