Step 1Peel the persimmons and cut each fruit into sixths.
Step 2In a heavy-bottom saucepan, combine the sugar and water and bring to a boil. Boil the mixture for a minute or so, then add the persimmons, along with 1 tablespoon lemon juice. Reduce the heat to a gentle simmer and cook until the persimmons are very soft, 1 to 1 ½ hours, stirring occasionally.
Step 3Strain the persimmons and place in a food processor, leaving the liquid in the pan. Pulse the persimmons until no large pieces remain. Return the persimmons back to the pan with the liquid. Alternatively, use an immersion blender to purée the persimmons in the pan.
Step 4Add the remaining lemon juice, ginger, cinnamon, nutmeg, salt, and cloves to the pan, stirring to combine. Cook at a gentle simmer, stirring frequently, until the mixture is thickened to a jam-like consistency, 10 to 15 minutes. This makes a scant 2 cups filling, more than enough for the cookies.
Step 1Heat the oven to 375 degrees.
Step 2In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a hand mixer, beat the butter, sugar and salt until light and fluffy, 3 to 5 minutes. Beat in the egg until fully incorporated.
Step 3Beat in the flour, then the ground nuts. If it becomes too difficult to add the flour or nuts using the mixer, continue stirring in by hand until combined.
Step 4Gently knead the dough, by hand, in the bowl until it holds together. Divide the dough in half. (If at any time the dough feels overly soft, cover and chill it until firm enough to handle.)
Step 5On a lightly floured surface, roll one-half of the dough until it is a little less than 1/4-inch thick. Cut out the dough using a 2-inch to 3-inch round cutter. Repeat with the remaining dough.
Step 6Using a 1-inch star-shaped cutter, cut out the center of half of the rounds (save the dough from the cutouts to make more rounds until all of the dough is used).
Step 7Place the cookies on a baking sheet and bake until the edges are very lightly browned, 6 to 8 minutes. Remove and set aside to cool.
Step 8Spread a thin layer of persimmon filling over each of the whole cookies, then top with one of the “window” cookies. For an additional decorative touch, sprinkle powdered sugar over the windows before topping.