Separate the egg yolks from the egg whites, placing the yolks in a large bowl, and the whites in the bowl of a stand mixer, or in a separate large bowl.
In a large bowl, using an electric hand mixer, combine sugar and egg yolks. Beat until the color is lighter and the mixture is bubbly. Stir in the milk, oil, and ube flavoring.
Sift the dry ingredients (cake flour and baking powder) into the the egg mixture. Combine cake flour and egg mixture until it is smooth and free of lumps.
In a separate large mixing bowl, or in the bowl of a stand mixer using a whip attachment, beat the egg whites until frothy. Add the cream of tartar and beat the egg whites until soft peaks form. Gradually add 1/4 cup of sugar to the egg whites and continue beating until stiff peaks form.
Using a rubber spatula, fold the beaten egg whites in three additions into the ube mixture until it is streak-free and uniform in color. Pour the ube cake batter over the flan layer in the pan. The pan will be almost full.
Place the cake in the preheated oven. Bake the ube flan cake for 25 minutes, or until the cake is set and springy.
Allow the cake to cool. To unmold, gently run a thin knife around the edges of the pan . Turn the cake over onto a serving plate.