Make the dough: In a small bowl combine the water and yeast and let proof for 5 minutes. In a food processor combine the flour, sugar and salt and process to combine -- about 30 seconds. Add the butter and process until mixture resembles fine meal. With the motor running add the yeast mixture and the milk, in a stream, and continue to process until a dough is formed. Transfer dough to a lightly floured surface and gently knead for 1 minute. Wrap in plastic and chill for at least 30 minutes, or overnight.
On a lightly floured surface pat the rested dough into a 10 by 4-inch rectangle. Roll the dough into a 14 by 9-inch rectangle.
In a bowl with an electric mixer beat the butter with remaining 2 tablespoons flour until it is pliable but still cold and firm. Between sheets of parchment or waxed paper roll the butter into a rectangle large enough to cover two-thirds of the dough with a one-inch border all around.
Using parchment paper, invert butter onto the bottom two-thirds of the dough, leaving a 1-inch border all around. Fold the top-third of the dough down over the top half of the butter. Fold the bottom third up over the top (like a letter) and pinch edges to seal.
Turn dough over, seam side down, give it one quarter turn and roll lengthwise into a 20 by 10-inch rectangle. Turn the dough over, seam side up, and fold like a letter. This is one turn. Chill dough, wrapped in plastic, for 30 minutes, or until firm.
Arrange dough on lightly floured surface, with folded edge to your left, and roll lengthwise into a 20 by 10-inch rectangle. Fold and chill as before. Repeat process two more times, giving the dough four turns in all. After the fourth turn chill dough for at least 3 hours, or overnight.
Line two baking sheets with parchment paper.
Form the croissants: On a lightly floured surface roll the dough into a 25 by 14 by 1/4-inch rectangle. Halve the dough lengthwise into two 7-inch strips. Using a pastry wheel or a long chef's knife, cut the strips into triangles, each with a base of 4 to 5 inches. Beginning with the base of each triangle, fold 1/4-inch of the dough over onto itself and begin rolling the triangle up to the point. Gently stretch the dough by pulling on the tip as you roll it. Repeat with the remaining triangles. If dough becomes soft, chill. Transfer the croissants to the baking sheets as they are formed spacing them 2 inches apart.
Make the glaze: in a bowl whisk together the egg and milk. Brush the croissants with the glaze and let rise, uncovered, in a cool place (about 65 to 75 degrees) for 1 1/2 to 2 hours.
Let croissants rise in a warmer place (about 75 to 80 degrees) for 1 1/2 hours more, or until almost doubled in size. Brush once more with the glaze and place on the center rack of a preheated 400 degree oven for 15 to 17 minutes, or until golden brown. Transfer to racks to cool.
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