Directions

1. Whisk together egg yolks and cornstarch in a small heatproof bowl until thickened and smooth, about 30 seconds; set aside next to stovetop.
2. Cut vanilla bean lengthwise; scrape out seeds using back of a paring knife. Bring granulated sugar, vanilla bean seeds, and 1 1/2 cups milk to a simmer in a medium saucepan over medium, whisking often. Remove from heat; gradually whisk about half of milk mixture into egg yolk mixture. Add warmed egg yolk mixture to remaining milk mixture in saucepan, whisking until smooth; heat over medium, whisking constantly, until mixture begins to thicken, 2 to 3 minutes. Once mixture thickens, whisk, pausing every few seconds to check for bubbles, about 1 minute. When mixture begins to bubble, set a timer, and whisk for 1 minute. Remove from heat; whisk in 1 tablespoon of the butter until smooth. Pour pastry cream through a fine mesh strainer placed over a heatproof medium bowl. Immediately place plastic wrap directly on surface to prevent a skin from forming; refrigerate until completely chilled, at least 2 hours or up to 24 hours.
3. While pastry cream is cooling, preheat oven to 400°F. Line a large unrimmed baking sheet using parchment paper; unfold 1 of the puff pastry sheets, and place on prepared baking sheet. Prick dough several times using tines of a fork to allow steam to escape. Cover dough using a second sheet of parchment paper. Place a large rimmed baking sheet directly over top of parchment-covered pastry sheet; spread pie weights or dried beans evenly over top baking sheet.
4. Bake until pastry is puffed, about 15 minutes. Remove top baking sheet and parchment paper, and continue baking until pastry is deep golden brown, 5 to 8 minutes. Remove pastry sheet from baking sheet; let cool completely on wire rack, about 30 minutes. While pastry sheet is cooling, repeat process with remaining puff pastry sheet.
5. Beat remaining 12 tablespoons (3/4 cup) butter with a stand mixer fitted with a paddle attachment on medium speed until fluffy and light, about 2 minutes. With mixer on medium speed, add cooled pastry cream, a few tablespoons at a time, beating until thoroughly incorporated, about 1 minute, stopping to scrape down sides of bowl as needed. Switch to a whisk attachment, and whip mixture on medium speed until smooth, soft, and airy, about 3 minutes. Transfer pastry cream mixture to a pastry bag fitted with a 1/2-inch round tip; refrigerate at least 1 hour or up to 4 hours.
6. Meanwhile, trim cooled puff pastry sheets to 8-inch squares. Cut each sheet into 12 (2 1/2- x 2-inch) rectangles. Place 8 rectangles on a wire rack placed inside a large rimmed baking sheet; set aside all pastry rectangles. Whisk together powdered sugar and remaining 2 tablespoons milk in a small bowl until smooth. Transfer about 1 tablespoon icing to a separate small bowl; whisk in cocoa until smooth, adding additional milk as needed, 1/4 teaspoon at a time, until consistency matches white icing. Transfer chocolate icing to a small piping bag; cut 1/4-inch opening in bag. Working with 1 rectangle at a time, spread about 2 teaspoons white icing over rectangle. Immediately pipe 4 lines of chocolate icing lengthwise over white icing. Lightly drag a wooden pick or skewer crosswise through icing, alternating directions to create a feathering or chevron pattern; clean wooden pick between each swipe. Repeat process with remaining 7 rectangles on wire rack; set aside.
7. For half of the plain pastry rectangles, pipe 12 (about 1/2-inch-high) dollops of pastry cream mixture in an even layer over each rectangle (3 rows of 4 dollops); place remaining plain pastry rectangles on top of pastry cream dollops, pressing lightly to adhere. Repeat piping process with remaining pastry cream mixture and plain rectangles. Top with reserved iced pastry rectangles (transfer by holding on to edges to avoid messing up design). Place assembled Mille Feuilles in an even layer on a platter. Refrigerate, uncovered, until set, at least 30 minutes or up to 4 hours.