Preheat oven to 350 F. Melt the cheddar cheese with the milk and vegetable oil in a medium saucepan over medium heat. Let cool.
In a large mixing bowl, whisk together the yolks and sugar until pale in color. Add the vanilla extract and the cooled cheese mixture and mix well. Combine all-purpose flour and baking powder and sift over the flour mixture. Stir the batter well until smooth.
In the bowl of a stand mixer (or use a hand-held mixer), beat the egg whites until foamy. Gradually add in the sugar as you continue to beat until the egg whites form firm peaks. Fold the egg whites to the yolk batter in three additions, gently combining until the batter is uniform in color.
Pour the batter into an ungreased 10-inch tube pan. Bake at 350 minutes for 30-35 minutes, or until the cake springs back to the touch and a skewer inserted at the center of the cake comes out clean. Cool the cake on a wire rack or upside down on top of an overturned mug.
To release the cake, loosen the sides with a knife and gently remove the loose bottom of the pan. Gently lift the cake and transfer the cake upside down on a cake serving plate.
In a mixing bowl, whip the butter with a mixer until light and fluffy. Add the icing sugar, milk and vanilla extract and beat until smooth. You may add a little more milk (one tsp at a time) to thin down the buttercream to your desired consistency.
Spread the buttercream on top of the cake and top with the shredded cheese.