Meanwhile, add milk, caster sugar, and butter to a saucepan. Cook over medium heat, stirring constantly, until the butter has melted. Bring the mixture to a simmer, then reduce the heat to medium-low and add the flour all at once. Continue stirring until the flour is fully incorporated and the mixture is smooth.
Turn off the heat and transfer the mixture to a clean mixing bowl. Let it cool for 3 to 5 minutes, then slowly whisk in the beaten egg, a little at a time. Once the mixture becomes thick, shiny, and smooth, stop whisking and transfer it to a piping bag using a spatula.
Preheat the oven to 200°C (410°F). While the oven is preheating, remove the pastry dough from the refrigerator and roll it out to about 2 mm (1/16 inch) thick. Use a 2-inch (5 cm) round cookie cutter to cut out circles from the dough.
Line a baking pan with parchment paper. Pipe 2-inch (5 cm) mounds onto the paper, spacing them about 3 inches (7 cm) apart. Top each mound with a piece of the cookie-like pastry dough.
Bake the pastries at 200°C (410°F) for 10 minutes. Then, reduce the oven temperature to 160°C (320°F) and continue baking for an additional 20 minutes, or until golden brown and crispy. Once baked, remove the pastries from the oven and transfer them to a cooling rack.