Make the dulce de leche: Empty the milk into the top of a double boiler or a heatproof bowl over a pan of simmering water. Cover with a tight fitting lid. Cook, stirring every 10 to 15 minutes, until the milk is thick and amber in color, about 5 hours. Remove from heat, and beat with a wooden spoon to smooth out. Transfer to a clean bowl, and refrigerate several hours, up to 3 days.

Line 2 baking sheets with parchment paper.

Sift together the flour and confectioners’ sugar. Add to a food processor, along with the butter and pulse until it resembles coarse meal, about 20 seconds. With the food processor running, pour the water in a slow stream and process just until dough comes together, about 20 seconds. Form the dough into two flattened disks, wrap well with plastic wrap, and refrigerate for 1 hour.

Preheat the oven to 350F. On a well floured work surface, roll out one disk into a 1/4 inch thickness. Using a 1 3/4 inch round cookie cutter, cut out rounds from the dough and transfer to the prepared baking sheet. Repeat with the other disk of dough. Gather up the scraps from both batches, reroll and cut. Sprinkle half of the rounds with sanding sugar. Bake until golden brown, about 15 mintues, rotating the sheets halfway through. Transfer to a wire rack to cool completely.

About 30 minutes before serving, spread 1 teaspoon of the cold dulce de leche on the bottom of the unsugared cookies. Place a sugared cookie on top to make sandwiches. Serve immediately.

Unfilled cookies can be stored in an airtight container at room temperature for 3 days.