Make the dough block: Pour the warm milk into a small bowl and whisk in 1 teaspoon of the sugar. Whisk in the yeast and set aside for 10 minutes.

In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, remaining 2 tablespoons of sugar, the salt, and cold butter pieces until the mixture resembles bread crumbs. Add the yeast mixture and the cold milk. Switch to the dough hook and mix on lowest speed for 1½ to 2 minutes. Knead the dough 3 to 5 times. Refrigerate for 30 to 60 minutes.

Make the butter block: Cut the butter into ½-inch pieces, toss with the flour, and refrigerate for 20 minutes. In a stand mixer fitted with the paddle attachment, beat the floured butter on medium speed, for 1 to 2 minutes, until the butter and flour form a smooth mass. Scrape the butter onto a piece of parchment paper or plastic wrap, and refrigerate while you roll out the dough.

Incorporate the butter into the dough: Dust the work surface with flour. Set the dough in the center. Roll the dough into a 15 by 12-inch rectangle with a short side parallel to the edge of your work surface. Spread the cold but pliable butter evenly over the top two sections of dough, leaving the bottom third empty and leaving a ½-inch border around the edges of the buttered sections.

Use a letter fold to encase the butter: Fold the empty bottom third up over the center third of the dough. Then fold the top third down over the center. Refrigerate for 1 hour before continuing with the second turn.

Dust the dough with flour and roll it into a 20 by 12-inch rectangle. Fold the dough using the book-fold method: Fold the two short edges into the center of the dough, leaving a ¼-inch crevice between them. Now fold one side over the other, as though you were closing a book. Wrap the dough in plastic and refrigerate for 1 hour.

Remove the dough from the refrigerator, dust with flour, and roll again into a 20 by 12-inch rectangle. Fold the dough using the letter-fold method: Visually divide the dough lengthwise into 3 equal, 5-inchwide sections. Fold the bottom third up over the center of the dough, and then fold the top third down over the center. Wrap and refrigerate for at least 2 hours and up to 24 hours.

Cut the dough: Roll the dough on a floured surface into a 26 by 14 by ¼-inchthick rectangle. Cut the rectangle in half lengthwise to form two pieces that each measure 26 by 7 inches. On each piece, use a ruler and paring knife to make nicks along the top edge of the dough every 4 inches. Along the bottom edge, measure 2 inches in from the left side and make a nick; then add a nick every 4 inches.

Now cut the dough into triangles: Line up your ruler with the top left corner and the first bottom nick. Cut along this line. Next, line up the ruler with the first nick on the top edge and the left corner bottom, and cut along that line Then cut a line from the first nick on top to the first nick on the bottom to form the second triangle. Continue cutting until the whole sheet.

Shape the dough: Line the baking sheets with parchment paper or silicone mats. Make a 1-inch vertical slit in the center of the bottom edge of each triangle. Now roll the entire triangle toward the tip.

Wash with egg and proof: In a small bowl, lightly beat the eggs and milk. Brush each croissant evenly with the egg wash. Allow the croissants to rise until they are nearly doubled in size, 1 to 2 hours.

Bake the croissants: Preheat the oven to 400°F and position a rack in the center. Chill the croissants in the freezer for 10 minutes or in the refrigerator for 15 minutes. Brush the croissants once more with the egg wash. Bake one baking sheet at a time, rotating it halfway through, for 17 to 22 minutes, until the croissants are a deep golden brown. Transfer the croissants to a rack to cool.