Dried-Apple Stack Cakes

Make filling:

Simmer all filling ingredients and a pinch of salt in a 4- to 5-quart heavy pot, uncovered, stirring occasionally and mashing apples with a potato masher as they soften, until a thick purée forms, about 1 1/2 hours. Cool completely.

Make cakes:

Preheat oven to 350°F with rack in middle. Butter muffin cups.

Stir together milk and vinegar and let stand 10 minutes to curdle.

Sift together flour, baking soda, salt, ginger, and cloves.

Beat butter and brown sugar with an electric mixer at medium speed until pale and fluffy, about 3 minutes. Beat in molasses and egg. At low speed, add flour mixture in 3 batches alternately with curdled milk, beginning and ending with flour mixture and mixing until each addition is just incorporated. Spread 1 heaping tablespoon batter in each muffin cup and bake just until a wooden pick inserted in center of a cake comes out clean with some crumbs adhering, 11 to 13 minutes. Cool 5 minutes in pans on racks, then invert onto racks to cool completely.

Assemble cakes:

Spread about 2 tablespoons filling on top of 1 cake and invert another cake over filling, then spread a heaping tablespoon filling on top. Make 11 more stacks (you will have about 2 cups filling left over). Transfer to an airtight container and chill at least 2 days for flavors to develop. Bring to room temperature before serving.