Preparation

Make the crust:

• Position an oven rack in the center of the oven and heat the oven to 350°F. Line the bottom and sides of a 9×13-inch baking pan with foil, allowing foil to overhang the long sides of the pan to act as handles for removing the cookie later. Lightly butter the foil.
• In a food processor, combine the flour, confectioners’ sugar, and salt. Process the ingredients briefly to combine, about 15 seconds. Scatter the cold butter pieces and the vanilla over the flour mixture and process, using short pulses, until the dough begins to form small clumps, 1 to 1-1/2 minutes. Turn the dough into the prepared pan. Using lightly floured fingertips, press the dough into the pan in a smooth, even layer. Bake until pale golden, especially around the edges, 22 to 25 minutes. Do not overbake or the crust will be hard and crispy. Transfer the pan to a cooling rack and lower the oven temperature to 325°F.

Make the filling:

• In a medium bowl, melt the chocolate, milk, and butter together over a pot of barely simmering water or in the microwave. Whisk until smooth and set aside to cool slightly.
• In a stand mixer fitted with a paddle attachment or in a large mixing bowl, using a hand-held electric mixer, beat the eggs, sugar, and Kahlúa on medium-high speed until foamy and lighter in color, 2 minutes. Reduce the speed to low and gradually add the chocolate mixture. Stop the mixer and scrape down the bowl and beater. Beat on medium speed until well blended, about 30 seconds.
• Pour the chocolate batter over the baked crust and spread evenly. Bake until the sides are slightly puffed and a toothpick inserted near the center comes out wet and gooey but not liquid, 30 to 35 minutes. Transfer the pan to a rack. As it cools, the center may sink a bit, leaving the edges slightly (about 1/2 inch) elevated. While the filling is still warm, use your fingertips to gently press the edges down to the level of the center, if necessary.
• When completely cool, cover with plastic and refrigerate until very cold, at least 12 hours or up to 2 days. To serve, using the foil as handles, lift the rectangle from the pan and set it on a cutting board. Tipping the rectangle, carefully peel away the foil. Using a hot knife, cut the rectangle lengthwise into 1-1/2-inch strips, wiping the blade clean before each cut. Cut each strip on alternating diagonals to make small triangles. Let sit at room temperature for about 5 minutes before serving.

Make Ahead Tips

You can bake these up to 1 month ahead: Wrap the cooled baking pan in heavy duty plastic wrap and freeze (no need to cut them into triangles first).The baked truffles can also be refrigerated, wrapped in plastic, for up to 2 days.