In a mixing bowl, combine the cake flour and sugar. Stir the mixture with a whisk to evenly distribute the sugar.
Add the cold, cubed butter to the flour mixture. Wear food handling gloves to prevent warming up the butter, use your fingertips to rub the butter into the flour until the mixture resembles coarse crumbs. The butter should be broken down into pea-sized pieces. Gradually add the beaten egg, 1 tablespoon at a time, while stirring with a hand. Use about 60% of a large egg (or as needed) until the dough can hold its shape when pinched with your fingers. Be cautious not to make the dough overly wet.
Gently use your hands to form the dough into a ball, avoid over-kneading the dough. Divide the dough in ⅔ and ⅓, flatten each piece of the dough into a disc. Wrap each dough disc in plastic wrap, and refrigerate for 1 hour or up to 1 day.
Preheat the oven to 175℃/350℉.
After chilling, take the larger dough out of the refrigerator and let it sit at room temperature for 10 minutes. On a nonstick silicone pastry mat (or floured surface), wrap a rolling pin with plastic food wrap, roll out the dough into a 11'' circle. Carefully line the dough sheet onto a 9'' quiche pan. Gently press the dough sheet to fit the mold. Trim off the excess dough on the edge, combine it with the other ⅓ of the dough in the refridgerator. Use a fork to gently press the tines into the bottom of the dough sheet. Space the holes about 1 inch apart. (This process allows steam to escape during baking, preventing the crust from puffing up in large bubbles.) Optionally, line the pie crust with parchment paper, and pour dried beans onto the parchment paper to weigh it down. (The added weight helps the crust maintain its shape during baking.)
Place the pie crust with the beans into the oven and bake at 175℃/350℉ for 18 minutes. Remove the pie crust from the oven, then carefully remove the parchment paper and the beans from the crust. Let the pie crust cool completely on countertop. (If you don't plan to use the pie crust within a day, it's best to wrap it tightly with plastic wrap and store it in the refrigerator for up to 3 days.)