FOR CAKE: Whisk dry ingredients and spices in a large bowl. Mix milk, eggs and vanilla in a 2-cup measuring cup.
Beat softened butter into dry ingredients, first on low, then medium, until mixture forms pebble-sized pieces. Add about 1/3 of the milk mixture and beat on low until smooth. Add remaining milk mixture in two stages; beat on medium speed until batter is just smooth. Add the sugar; beat until just incorporated, about 30 seconds. Pour batter into a greased and lightly floured 13x9" cake pan.
Bake in a pre-heated 350-degree oven on middle rack until a toothpick inserted into the cake's center comes out clean, about 40 minutes. Set pan on a wire rack; let cool for 5 minutes. Run a knife around the pan perimeter and turn cake onto rack. Let cool.
FOR FROSTING: Cream butter and gradually add half the sugar, blending well. Beat in 2 Tbsp. cream and the vanilla. Gradually blend in remaining sugar. Add enough of the cream to make it a good spreading consistency. Frost the cake after it has cooled.