Stack cake, an old Appalachian dessert, is often made with an apple-butter filling, but we chose dried-plum butter instead.
Dried-plum butter: Bring dried plums, orange juice, lemon zest and juice, and 2 cups water to a boil in a covered medium saucepan. Reduce heat to low and simmer, stirring occasionally, until fruit is beginning to break down, 35 to 40 minutes. Stir in 1/4 cup water, brown sugar, and salt, mashing mixture with the back of a spoon until jam-like. Transfer to a medium bowl; let cool completely. Dried-plum butter can be stored in refrigerator, covered, up to 1 week.
Cookies: Whisk together flour, baking powder, salt, cardamom, and baking soda in a medium bowl. Beat butter with sugars on medium-high speed until light and fluffy, about 2 minutes. Beat in egg. Reduce speed to low and beat in flour mixture in 2 batches, alternating with milk, until combined. Divide dough into 9 pieces. Shape into disks, wrap each in plastic, and refrigerate at least 1 hour and up to 1 day.
Preheat oven to 350 degrees. Working with one dough piece at a time (keep the rest in refrigerator), roll out on a lightly floured surface with a floured rolling pin into a roughly 8 1/2-inch round. Using a paring knife and a cake pan or plate as a guide, cut dough into an 8-inch round. Transfer to a parchment-lined baking sheet; refrigerate while rolling and cutting remaining dough pieces, placing 2 rounds on each sheet.
Bake cookies in batches until golden and firm in center, 10 to 12 minutes. Let cool completely on sheets on wire racks.
Place 1 cookie on a cake plate and spread with 1/2 cup dried-plum butter, leaving a 1/4-inch border. Top with another cookie and spread with another 1/2 cup dried-plum butter. Continue stacking and spreading until all 9 cookies have been used. (Do not spread filling on top layer.) Wrap in plastic; refrigerate at least 12 hours and up to 1 day.
Serve, dolloped with whipped cream and sprinkled with cinnamon.