Preheat oven to 350°.
To prepare the gingerbread, coat a 9-inch round cake pan with cooking spray; line bottom of pan with parchment paper. Coat paper with cooking spray; set aside.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and next 6 ingredients (through cloves) in a bowl, stirring with a whisk. Combine 3/4 cup sugar and 1/2 cup butter in a large bowl; beat with a mixer at medium speed until light and fluffy. Add egg; beat until blended. Beat in molasses and fresh ginger. Add flour mixture and beer alternately to sugar mixture, beginning and ending with flour mixture. Pour batter into prepared pan. Bake at 350° for 40 minutes or until a wooden pick inserted into center comes out clean. Cool in pan 10 minutes on a wire rack; remove from pan. Cool completely on wire rack.
To prepare sauce, combine 1/2 cup sugar, 1/2 cup water, and cornstarch in a medium saucepan; bring to a boil. Cook 1 minute, stirring constantly. Remove from heat. Add 1 tablespoon butter, stirring until smooth. Stir in rind, juice, and vanilla. Cool to room temperature. Serve with gingerbread.