Heat oven to 400ºF. Line an 18-by-13-inch baking sheet with parchment paper. Have another 18-by-13-inch baking sheet and piece of parchment on hand.
Roll out the puff pastry to a 12-by-15-inch rectangle. Cut in three equal rectangles, about 12 by 5 inches each. Dock all over with a fork to suppress the puff pastry's rise in the oven. Transfer the pieces to the prepared baking sheet, cover with the other sheet of parchment, and top with the second baking sheet. Bake until the flour structure has set and the pastry looks light golden and flaky, 10 to 12 minutes. Remove the top baking sheet and piece of parchment and return to the oven until golden brown, 4 to 5 minutes more. Cool completely on a wire rack.
Meanwhile, combine the sugar, salt, vanilla, and 2 1/2 cups of the milk in a large saucepan. Cook over medium-high heat until bubbles form around the edges. While the milk is warming, whisk together the cornstarch, flour, eggs, and remaining 1/2 cup milk in a large bowl.
Temper the egg mixture with the hot milk mixture, adding a ladleful of the hot milk to the egg mixture at a time, whisking the whole time. Once you've added two-thirds of the milk mixture to the egg mixture, pour it back into the pan and cook on medium heat, whisking constantly. Once the mixture thickens and comes to boil, stir vigorously and count to 60. Remove from the heat.
Pass the pastry cream through a fine-mesh sieve into a bowl, then stir in the butter and Scotch. Cool over an ice bath, stirring occasionally, until the pastry cream reaches room temperature. Transfer the pastry cream to a pastry bag fitted with a medium round tip. If not using immediately, cover with plastic wrap, pressing the plastic wrap onto the surface of the pastry cream to keep a skin from forming.
Whip the cream in a medium bowl with a hand mixer or balloon whisk until medium-stiff peaks form. Put the whipped cream in a separate pastry bag with a medium round tip.
Trim the cooled puff pastry carefully with a serrated bread knife to produce three identical pieces. Then, slice the long rectangle pieces crosswise into 2-inch pieces.
Line up a third of the pieces to form the first layer of the mille-feuille, like a puzzle. Pipe dots of pastry cream on the first layer. Top with the next layer of pastry. Pipe dots of whipped cream on the second layer of pastry. Line up the pieces of pastry for the final layer on a baking sheet. Dust the top with the confectioners' sugar, covering the pastry completely. Torch the top, being careful not to burn the sugar. Dust with a second layer of confectioner’s sugar and torch again. Sprinkle with the smoked salt, and allow the pastry to cool before handling, about 3 minutes. Top the mille-feuille with the final layer of bruleed pastry. Serve immediately.