Heat oven to 350 degrees. Grease an 8-inch springform pan. For cake: Sift together the flour, baking soda and salt; set aside. Put 1 cup root beer in a small saucepan; boil, uncovered, until it is reduced to 1/2 cup, about 5 minutes. Cool to lukewarm. Beat brown sugar and butter in large bowl of an electric mixer on high speed until light and fluffy, about 2 minutes. Add the eggs, one at a time, mixing well after each addition. Stop the mixer and add the reduced root beer and vanilla. Mix just to combine. Fold in the dry ingredients with rubber spatula. Transfer batter to prepared pan. Bake until a toothpick inserted in the center comes out clean, 45 to 50 minute. Cool 5 minutes. For frosting: Poke holes all over top of cake with a toothpick. Brush with about 3 tablespoons of the root beer. Remove the sides from the pan and cool cake completely. Mix remaining 3 tablespoons root beer, confectioner's sugar, butter and vanilla in a small bowl to make a thin frosting. Spread over top of cooled cake, letting it drip down the side. ----