Bring prunes, ¾ cup granulated sugar, the cinnamon, cloves, and ¾ cup water to a simmer in a 2-qt. saucepan over medium-high; cook until prunes are soft and the syrup has thickened, about 20 minutes. Let cool slightly and then transfer to a food processor; purée until smooth and set purée aside.

Whisk flour, baking powder, and cardamom in a bowl; set aside. In another bowl, and using an electric hand mixer, cream remaining granulated sugar and the butter until fluffy. Add eggs one at a time, beating well after each addition. With the motor running, slowly add dry ingredients and the milk until dough forms. Transfer dough to a lightly floured work surface; knead until smooth, 1–2 minutes. Shape dough into a ball and cover with plastic wrap; chill 1 hour.

Heat oven to 375°. Divide dough into 6 balls. On a lightly floured surface, and working with 1 ball of dough at a time, roll dough into a 7" disk, about ⅙" thick. Place disks on parchment paper-lined baking sheets; bake until golden and cooked through, about 25 minutes; let cool slightly. To assemble, place 1 disk on a baking sheet fitted with a wire rack. Spread with ⅓ cup reserved prune purée. Repeat layering process with remaining disks and purée, ending with a disk. Whisk confectioners' sugar and coffee in a bowl until smooth and pour over cake; chill until icing is set, about 10 minutes. Transfer cake to a cake stand or plate to serve.