The base of these cupcakes is a springy chocolate cake called "wacky cake." Made with no butter, eggs or milk and stirred together in one bowl, it was popular during World War II when rations were low. Along the way, someone added a rich cheesecake filling and black bottom cupcakes were born.
Position a rack in the center of the oven and preheat to 350 degrees F. Line a 12-cup muffin tin with liners. Beat together the cream cheese, egg, 1/3 cup of the granulated sugar and 1/2 teaspoon of the vanilla extract in a medium bowl on medium speed with an electric mixer until smooth and no lumps remain, 1 to 2 minutes. Set aside.
Whisk the flour, cocoa powder, brown sugar, baking soda, salt and remaining 2/3 cup granulated sugar in a large bowl. Add the oil, the remaining teaspoon vanilla extract and 1 cup water and mix until combined. Let the batter rest 10 minutes at room temperature, then stir in the vinegar.
Divide the chocolate batter evenly among the prepared muffin cups, filling each cup about two-thirds full. Drop about 2 tablespoons of the cream cheese filling in the center of each cupcake.
Bake until the cream cheese filling has set and a toothpick inserted into the chocolate cupcake comes out clean, about 20 minutes.
Cool in the tin 5 minutes and then use a butter knife or small offset spatula to carefully remove the cupcakes to a wire rack. Cool the cupcakes completely before serving, about 30 minutes.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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