Whisk the water, yeast, and 1 tablespoon sugar together in a large bowl. Cover and allow to rest until foamy on top, about 5 minutes. Whisk in remaining sugar, the milk, egg, and salt. Once these wet ingredients are mixed together, lightly cover and set the bowl aside as you work on the next step.
Cut the cold butter into ¼ inch slices and add to a food processor. Top with 2 ½ cups flour. Pulse the mixture 12-15 times, until butter is crumbled into pea-size bits.
Pour the flour mixture into the wet yeast mixture. Very gently fold everything together using a rubber spatula or wooden spoon. Fold just until the dry ingredients are moistened.
Turn the sticky dough out onto a large piece of plastic wrap.
1st Refrigeration: Wrap the dough up tightly and refrigerate for at least 4 hours and up to 48 hours.
Roll and Fold: Take the dough out of the refrigerator. Very generously flour a work surface. The dough is very sticky, so make sure you have more flour nearby as you roll and fold. Using the palm of your hands, gently flatten the dough into a small square. Using a rolling pin, roll out into a 15x8-inch rectangle. When needed, flour the work surface and dough as you are rolling. Fold the dough into thirds as if it were a letter. Turn it clockwise and roll it out into a 15-inch long rectangle again. Then, fold into thirds again. Turn it clockwise. Repeat rolling and folding 1 more time for a total of 3 times.
2nd Refrigeration: Wrap up tightly and refrigerate for at least 1 hour and up to 24 hours.