Remove the milk and butter from the refrigerator. Cut the butter into tablespoon-sized cubes and set aside.
Spray a large mixing bowl with cooking spray and set aside.
In the work bowl of a stand mixer, add 3 ½ cups of flour. Add the sugar and salt on one side of the bowl. Stir the dry ingredients with a whisk or the dough hook to combine.
If using Instant Yeast, add the yeast on the other side of the bowl away from the salt and stir to combine. If using active dry yeast, sprinkle the yeast over room temperature water in a liquid measuring cup. Allow the yeast to proof and get foamy for about 3 to 5 minutes.
Mix the dough: Pour the water and milk over the flour mixture. Mix the dough on medium-high speed until it is smooth and no longer sticky, about 8 to 10 minutes. If the dough is very sticky, add more flour a tablespoon at a time. If it is very dry, add more water, a teaspoon at a time.
Rest the dough: Remove the dough from the mixer and gently form it into a smooth ball. Place it in the prepared mixing bowl and cover with plastic wrap. Rest the dough in a warm place for at least 30 minutes.
Butter Square: Wipe out the bowl of the mixer and attach the paddle attachment. Add the cubed butter and 2 tablespoons of flour and beat until smooth and creamy. Do not overmix.
Shape and chill the butter square: Scrape the butter onto the middle of a long piece of plastic wrap. Using the plastic wrap, shape the butter into a flat square approximately 6 inches by 6 inches wide and 1 inch thick. Wrap the butter completely in the wrap and chill in the refrigerator while the dough is proofing.
Shape the dough: Once the dough has proofed, pour it onto a generously floured work surface and sprinkle the top lightly with flour. Roll the dough into a square, approximately 10 inches by 10 inches.
Place the butter square: Remove the chilled butter from the refrigerator and place it diagonally in the center of the dough. Fold the four corners of the dough over the butter, tightly pinching each of the seams to close tightly.
Refrigerate the dough: Place the dough packet on a baking sheet, cover with plastic wrap, and refrigerate for 60 minutes.
Roll and fold the dough: Place the chilled dough packet on a well-floured surface. Roll into a 16-inch by 8-inch rectangle. Brush off the excess flour. Fold each side into the center of the dough like folding a letter into thirds. Refrigerate again for at least 60 minutes.
Repeat the rolling, folding, and refrigerating process two more times for a total of three turns. After the final turn, wrap the dough in plastic wrap and refrigerate overnight.
Remove the chilled dough from the plastic wrap and place it on a lightly floured surface. Cut the rectangle in half. Roll each half into a 13-inch by 10-inch rectangle. Trim the edges to make them straight.
Cut croissant triangles: Using a pizza cutter or a sharp knife, cut the dough into triangles, each 4 inches by 9 inches.
Start with the wide end of the triangle and roll up to the point.
Repeat the steps for the other half of the dough.
Freezing note: At this point, you can freeze the croissants before the final proof.
Final proof: Place the shaped croissants on prepared baking sheets. Cover with a proofing bag or plastic wrap and proof in a warm place until doubled in size, about 60 to 90 minutes.
Preheat the oven to 375°F (350°F if using a convection oven). Brush the beaten egg on each croissant and bake for 20 minutes, rotating the pans halfway through, until golden.
Transfer the croissants to a wire rack to cool completely. Enjoy!