Simmer prunes, water, and sugar in a saucepan over medium heat until very soft, 12–15 minutes.
Stir in cardamom, cinnamon, salt, lemon zest, vanilla, and almond extract. Mash or puree until thick. Cool completely.
Heat oven to 350°F. Line baking sheets with parchment.
Whisk flour, baking powder, cardamom, and salt in a bowl.
Cream butter and sugar until light. Mix in sour cream and vanilla until smooth.
Add dry ingredients to the bowl and mix just until a soft dough forms. Chill 30 minutes.
Divide dough into 6 equal pieces. Lightly flour parchment and roll each into an 8-inch round.
Dock each round with a fork. Transfer to sheets and bake 8–10 minutes until pale gold at the edges.
Cool layers on racks. Keep unbaked rounds chilled while others bake.
Assemble on a board: layer one cake round, then spread prune filling thinly. Repeat, finishing with a plain top.
Wrap the stack tightly and press with a light weight. Rest at room temperature overnight.
Unwrap, dust with powdered sugar, and slice with a sharp serrated knife.