Place the flour, powdered sugar, baking powder, and salt in the bowl of a food processor and pulse a few times to combine. Add the butter and vanilla extract and process for about 30 seconds, until the dough comes together and rides around the sides of the bowl.
Form the dough into a 7-inch round log and wrap in a piece of waxed paper. Chill in the refrigerator for at least 6 hours, until firm. After 2 hours, remove the dough from the refrigerator and roll on a flat surface to form a smooth, circular log. Return to the refrigerator.
Preheat the oven to 350°F and line a few baking sheets with parchment paper. Remove the waxed paper from the dough and cut the dough into very thin slices, about ¼-inch thick. Arrange the cookies on the baking sheets. Bake for about 15 minutes, just until golden brown around the edges. Remove from the oven and let cool for 5 minutes. Transfer the cookies to a wire rack and let cool completely.
Increase the oven temperature to 425°F. Pour the condensed milk into a small, shallow baking dish and cover tightly with aluminum foil. Place the baking dish in a larger roasting pan. Fill the roasting pan with enough water to come halfway up the sides of the baking dish. Roast for 1 hour and 15 minutes, stirring once while cooking, until the milk turns a light golden color. Remove the baking dish from the water and transfer the dulce de leche to a small bowl. Let cool completely, then stir in the pecans.
To assemble the cookies, spread a generous amount of dulce de leche on the flat sides of half the shortbread cookies. Place the remaining cookies on top and press gently to secure.