Step 1Heat the oven to 350 degrees.
Step 2In the bowl of a stand mixer using the paddle attachment, or in a medium bowl using an electric mixer, cream together the butter and granulated sugar until smooth, then slowly stir in the flour until combined.
Step 3On a well-floured surface, roll the dough out to one-fourth-inch thick and cut out cookies using a 2-inch round cookie cutter.
Step 4Place the dough on a parchment- or Silpat-lined cookie sheet and bake until set, 8 to 10 minutes (do not let them brown). Cool a few minutes, then place the cookies on a rack to cool completely.
Step 5While the cookies are cooling, make the icing by mixing together the milk and powdered sugar to the consistency of thick syrup (the icing will have the consistency of school glue and should not run off the cookie when spread over); add a little more sugar to thicken or milk to thin, as needed.
Step 6Sandwich the cookies, placing a generous teaspoon of jam between the cookies. Spread the icing over the cookies and sprinkle with coconut; set aside until the icing is hardened.