Make the Pastry Dough: 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 1/4 inch slices and add to a food processor or blender. Top with 2 and 1/2 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, parchment paper, aluminum foil, or into any container you can tightly cover.
1st Refrigeration: Wrap the dough/cover up tightly and refrigerate for at least 4 hours and up to 48 hours.
Roll & 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 business 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/seal tightly and refrigerate for at least 1 hour and up to 24 hours.
Line two large rimmed baking sheets with parchment paper or silicone baking mats.
Take the dough out of the refrigerator and cut it in half. Wrap 1 half up and keep refrigerated as you work with the first half.
Shape into rounds: Cut the first half of dough into 8 even pieces. This will be about 1/4 cup of dough per pastry. Roll each into balls. Flatten each into a 2.5 inch circle. Use your fingers to create a lip around the edges. Press the center down to flatten the center as much as you can so you can fit the filling inside. (Center puffs up as it bakes.) Arrange pastries 3 inches apart on a lined baking sheet. Repeat with second half of dough.
Spoon 2 teaspoons of fruity filling or 1 tablespoon of cheese filling inside each.
Egg Wash: Whisk the egg wash ingredients together. Brush on the edges of each shaped pastry.
Chill the shaped pastries in the refrigerator, covered or uncovered, for at least 15 minutes and up to 1 hour.
Preheat oven to 400°F (204°C).
Bake for 19-22 minutes or until golden brown around the edges. Some butter may leak from the dough, that's completely normal and expected.
Remove baked pastries from the oven. Cool for at least 5 minutes before icing/serving.
Make the icing: Whisk the icing ingredients together. If you want a thicker icing, whisk in more confectioners’ sugar. If you want a thinner icing, whisk in more milk or cream. Drizzle over warm pastries and serve.
Cover leftover iced or un-iced pastries and store at room temperature for 1 day or in the refrigerator for up to 5 days. Or you can freeze them for up to 3 months. Thaw before serving. Before enjoying, feel free to reheat leftover iced or un-iced pastries in the microwave for a few seconds until warmed.