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Sprinkle 3 Tbsp flour over butter and blend together in a bowl. On a length of foil, fashion a 6" square of soft butter; fold over the sides of the foil to enclose. Place in the refrigerator to chill for 2 to 3 hours. While the butter is chilling, prepare the dough. You can mix by hand in a large mixing bowl but I suggest using an electric mixer, blend 2 cups of the flour with salt and sugar. Dissolve yeast in warm water and add it, the warmed milk and half-and-half to the flour mixture. Stir with a wooden spoon or use the flat blade of your electric mixer to thoroughly blend the dough, about 2 minutes. It should be batter-like. Stir in additional flour, 1/4 cup at a time, to make a soft but not sticky dough (it will stiffen when chilled.) and you may need to add more flour until it is no longer sticky. Knead by hand or with a dough hook for 5 minutes to form a solid ball, cleaning the sides of the bowl. DO NOT OVER KNEAD! Cover the bowl with plastic wrap and place in the refrigerator for at least 1 hr.
The butter and the dough need to be about the same temperature. The butter should bend, but not crack and break (too cold) or be mushy and oily (too warm) when you slightly bend it. This is the same for the dough. You are going to be incorporating the two together so they need to be pliable. You may need to take the butter out of the fridge before the dough. Place the dough on a floured work surface and with the hands press it into a 10" square. Unwrap the block of butter and lay the block diagonally on the dough. Bring each point of dough into the center, overlapping the edges at least 1". Press the dough into a neat package. With a heavy rolling pin (I use a marble one), roll the dough into a rectangle, approximately 8" x 18". This dimension is not critical. Warning: If the butter seems to be breaking into small pieces under the dough rather than remaining solid, allow the dough/butter to warm a few minutes. But if the butter softens, becomes sticky, and oozes while making the turns, put the dough back into the refrigerator for several minutes. Fold the length of dough into thirds, as for a letter. Turn so that the open ends are at twelve and six o'clock. Roll again into a rectangle. This time, fold both ends into the middle and then close, as one would a book. The dough will now be in 4 layers. Wrap the package of dough in a cloth (an old tea towel is good) that has been soaked in cold water and wrung dry. Place the wrapped dough in the refrigerator to relax and chill for 1 or 2 hours. Remove the dough from the refrigerator and place on the floured work surface. Unwrap, roll out, and fold in thirds, as for a letter. This is the final turn before it is rolled out and cut into croissants. Dampen cloth again and wrap loosely around the dough. Place the package in a plastic bag so moisture will be retained (not pulled out of the cloth). Leave in the refrigerator 4 to 6 hours or overnight.
Mix together the egg and 1 Tbsp of water. Sprinkle work surface with flour. Roll the dough until it is a generous 10"-x-38" rectangle, and, most importantly, about 1/4" thick. Make sure this is a precise dimension, because it determines the size and texture of the croissants. Trim irregularities to make the strip uniform in width. Cut the strip lengthwise to make two 5" pieces. Mark the strip into triangles, 5" wide on the bottom. Separate the triangles, place them on a parchment lined baking sheet, and chill for 15 to 20 minutes. Roll the dough into the traditional croissant shape, by rolling the triangle from the bottom to the point. Place the croissants on a parchment covered baking sheet and allow them to rise for 1 to 2 hours, in which they will double in volume. Preheat the oven to 425°F (220°C). Bake the croissants for 22 to 25 minutes. Allow them to cool on a rack before serving.
Makes 24 to 30 croissants.