Blend half stick (¼ cup) softened butter into 3¾ cups of the flour, the sugar, salt, and yeast in a large bowl; add milk; stir until dough forms. Transfer dough to a lightly floured surface; knead for five minutes; sprinkle flour as needed to prevent dough from sticking to surface. Press dough into a rectangle ~5×7 inches; cover with plastic wrap; refrigerate 30 minutes.
Roll chilled dough into a ~15×10-inch rectangle, as precisely as possible; wrap dough with plastic wrap; refrigerate 4-24 hours
On a lightly floured surface, roll out the chilled dough into a rectangle (~9×15-inches); place butter block in the center of the dough and fold each end of the dough over it; seal edges with your fingers. Roll out again to ~9_15-inch rectangle; fold the dough into thirds, like a letter; turn the resulting rectangle 90° (quarter turn). Repeat this process two more times.
Wrap the final folded dough with plastic wrap; refrigerate 4-24 hours.
On a lightly floured surface, roll out the chilled dough to an 8×20-inch rectangle. Using a pizza cutter or sharp knife, slice the dough in half vertically, yielding 2 4×20-inch rectangles; cut each into fourths, yielding 8 4×5-inch rectangles; cut each of these rectangles diagonally, yielding 16 triangles.
Using your fingers or a rolling pin, stretch a triangle to ~8 inches long; cut a small notch at the wide end of the triangle; starting with the wide end, tightly roll up, making sure the tip is underneath; bend the ends in towards each other, forming a crescent shape; repeat this process for all triangles.
Place the shaped croissants on 2 parchment lined baking sheets; rest at room temperature one hour; cover and refrigerate 1-12 hours.
Preheat oven to 400°F; whisk egg and 2 tablespoons milk to frothy; brush each croissant lightly with egg wash.
Bake ~20 minutes, until golden brown; rotate pan(s) halfway through baking.
