Activate the Yeast: In a small mixing bowl, combine lukewarm water, lukewarm milk, yeast, and 1 tablespoon of sugar. Stir until the yeast is fully dissolved. Let the mixture sit for 10-20 minutes, or until it becomes foamy.
Prepare the Dough: Pour the yeast mixture into the bread machine container. Add the egg yolk, 1/4 cup sugar, butter, salt, flour and vanilla pudding. Select the "Dough/Knead and First Rise" cycle, then press Start.
Divide the Dough: Once the cycle is complete, transfer the dough to a lightly floured surface. Divide it in half and roll each portion into a circle, about 18-20 inches in diameter. Cut each circle into 8 equal wedges.
Fill and Roll: Place your desired filling at the wide end of each wedge. Gently roll the wedges from the wide end to the tip, forming crescent shapes. Place them point-side down on a cookie sheet lined with parchment paper or an ungreased baking sheet.
Let the Dough Rise: Cover the pastries with a clean kitchen towel and let them rise in a warm place for 1 hour. While the dough is rising, preheat your oven to 370°F (190°C).
Apply the Egg Wash: In a small bowl, whisk together 1 egg white and 2 tablespoons of water until smooth. Using a pastry brush, lightly coat the surface of the pastries with the egg wash to ensure a shiny, golden finish after baking.
Bake: Bake the pastries in the preheated oven for about 15 minutes, or until they are golden brown. Remove from the oven and let them cool slightly before serving.