Beat the butter, confectioners' sugar, granulated sugar and orange zest in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the vanilla and beat until smooth. Combine the flour and salt in a medium bowl, then add to the butter mixture and beat until incorporated. Scrape down the sides of the bowl, then mix for an additional 20 seconds. Wrap the dough in plastic wrap and refrigerate for 30 minutes.
On a lightly floured surface, roll out the dough until it’s 1/4 inch thick. Using a 3-inch round cookie cutter, punch out as many circles as you can. Arrange the circles on 2 parchment-lined baking sheets. Gather the scraps, reroll and cut out more circles. If the dough starts to get soft, return to the refrigerator.
Preheat the oven to 325˚ F. Using a 3-inch cookie stamp, mark each cookie. Refrigerate at least 15 minutes.
Bake the cookies until lightly golden around the edges, about 10 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Spoon the dulce de leche into a pastry bag and snip a corner. Pipe the dulce de leche on half of the cookies, making sure to pipe the filling on the side without the stamp. Top each with another cookie, stamp-side up.