Make the pastry: In the bowl of a stand mixer fitted with a dough hook, add ½ cup cold water, the flour, powdered sugar, and 2 tablespoons butter. Mix on low speed until a smooth dough forms, about 3 minutes. Transfer the dough to a work surface, and press it into an 8-inch square. Wrap tightly in plastic wrap, and refrigerate to chill, at least 15 minutes.
Place the remaining 13 tablespoons butter between two large sheets of parchment paper. Use a rolling pin to flatten the butter to a 4-inch square, then refrigerate the butter block until it is chilled but still pliable, 8–10 minutes.
Lightly flour a work surface and unwrap the dough and the butter block. Place the butter angled atop the dough in a diamond position, and fold the corners of the dough in to meet at the center. Pinch the seams to enclose the butter in dough, pressing out any air.
Roll the dough out to a 6-by-12-inch rectangle. Lift one short side over the other to fold the dough into thirds. Wrap the dough in plastic wrap, and refrigerate for 15 minutes. Repeat the rolling and folding 3 more times, chilling the dough for 15 minutes between each turn.
Lightly flour the work surface, and lay the unwrapped dough on top. Roll it out to a 9-by-12-inch rectangle. Brush any residual flour from the surface of the dough, then brush the surface lightly with cold water. Starting with one of the longer sides, roll the dough into a tight cylinder. Wrap it in plastic wrap, and refrigerate for at least 6 hours or up to 3 days.
Place a rack in the top of the oven, and preheat to 450°F. Forty-five minutes before you plan to bake, unwrap the dough roll, and use a sharp knife to cut it crosswise into 22 half-inch-wide slices. Press each into a 2¾-inch mold; the edges should barely and evenly come up over the rim. Transfer to a large baking sheet, and refrigerate for 15 minutes.
Meanwhile, mix the filling: In a large bowl, whisk the yolks and sugar together, then whisk in the milk, followed by the cream. Strain through a fine strainer, then fill the shells just shy of the edges (about 2 tablespoons per tart). Bake until the custard is set and browned in spots, 20–25 minutes. Let cool slightly before unmolding. Serve warm or at room temperature.