Step 1

In a medium bowl, combine the flour, water and yeast and mix well with a wooden spoon. Cover with plastic wrap and refrigerate overnight.

Step 2

Let the sponge stand at room temperature for 30 minutes. Lightly grease a large bowl with olive oil.

Step 3

In a small saucepan, heat the milk until lukewarm. In the bowl of a stand mixer fitted with the dough hook, combine both of the flours with the sugar, salt and yeast. Scrape in the sponge. At low speed, drizzle in the lukewarm milk and mix just until the dough comes together. Let stand for 10 minutes.

Step 4

Knead the dough at low speed until it forms a ball, about 3 minutes. Transfer to the prepared bowl, cover with plastic wrap and let stand at room temperature for 30 minutes. Punch down and refrigerate for 3 hours.

Step 5

Meanwhile, on a sheet of parchment paper, arrange the sticks of butter side by side so they touch. Cover with another sheet of parchment. Using a rolling pin, pound the butter flat and roll out to a 10-by-12-inch rectangle. Refrigerate until ready to use.

Step 6

Remove the butter from the refrigerator and let stand at room temperature for about 10 minutes, until soft and pliable—the butter should yield gently when you press it with your finger and hold the indent, like a ripe avocado. On a lightly floured surface, using a lightly floured rolling pin, roll out the dough to a 12-by-28-inch rectangle. If the dough springs back, cover it with a kitchen towel and let rest for 5 minutes before rolling. Arrange the butter in the center of the dough so the short sides are parallel. Fold over the 2 short sides of the dough so they meet in the center; pinch together to seal. Pinch the long sides of the dough together to completely seal in the butter. Turn the dough so that a long side is facing you and roll out to a 12-by-28-inch rectangle. Starting from the bottom, fold the dough into thirds like a letter. Brush off any excess flour and wrap the dough in plastic. Chill until firm but pliable, about 1 hour.

Step 7

Arrange the dough on a lightly floured surface with a short side facing you. Using a lightly floured rolling pin, roll out the dough to a 12-by-28-inch rectangle. Fold into thirds like a letter, wrap in plastic and freeze overnight. Before baking, let thaw in the refrigerator until semi-firm and pliable, about 3 hours.

Step 8

Line 3 rimmed baking sheets with parchment paper. On a lightly floured surface, using a lightly floured rolling pin, roll out the layered dough to a 13-by-36-inch rectangle. Using a pizza cutter or sharp knife, cut the dough crosswise to form nine 13-by-4-inch strips. Halve each strip lengthwise on the diagonal to make 18 long triangles. Starting at the shortest side, roll each triangle into a crescent shape. Arrange the croissants on the prepared sheets, tip side down and spaced 3 inches apart. Cover loosely with plastic wrap and let rise at room temperature until doubled in size, 1 to 2 hours.

Step 9

Preheat the oven to 400°. In a small bowl, beat the eggs with 1 tablespoon of water. Brush the croissants with the egg wash and bake until deep golden brown and crisp, 20 to 23 minutes. Transfer the croissants to a rack to cool.