MAKE THE STARTER: About 8 to 12 hours before mixing your dough, combine the starter ingredients: starter, water, and flour in a bowl. Mix well, cover, and leave at room temperature in a warm place to allow the yeast to activate. It will become bubbly and double in volume.
WARM THE MILK: Heat the milk until it reaches a temperature of 110°F to 120°F.
WEIGH THE INGREDIENTS: Making sure to tare after each addition. combine 70 grams of warm milk, 100 grams of warm water, 200 grams of starter, and the sugar in a mixing bowl Add the all-purpose flour and whole-wheat flour.
MIX: Mix the ingredients together until a shaggy dough forms, then add the salt and the remaining 10 grams of water. Mix to combine. Add the unsalted butter and knead into the dough.
KNEAD: Turn the dough out onto a work surface and knead for 10 to 15 minutes by hand or 3 to 7 minutes with a dough hook in a stand mixer, until the dough is smooth, is no longer sticky, and releases easily from the bowl or work surface.
BULK FERMENT: Cover the dough and leave it to ferment for 4 to 8 hours, until it has doubled in volume.
PREPARE THE BAKING SHEETS: Line two flat baking sheets with parchment paper and set aside.
PREPARE THE MOLDS: Cut 8 squares of aluminum foil. Crumple and fold each foil sheet into a cone shape about 4½ inches tall with a top that is about 2 to 2½ inches across.
DIVIDE: Divide the dough into 8 equal pieces (about 98 grams each).
SHAPE: Transfer the dough to a work surface. Roll a piece of dough under your hands until it becomes a 25-inch rope. Start at the top of the foil cone and wrap the dough around the outside. Pinch the first coil to seal the loop, then wrap the dough around the cone, keeping the coiled layers tight together. Once you get to the bottom, pinch the ends to close the tip of the cone shape. (If you run out of dough, you can gently push the coiled layers down the paper-lined baking sheets. Repeat with the remaining pieces of dough. There should be 4 shaped rolls on each baking sheet, spaced at least 3 inches apart.
PROOF: Cover and proof in a warm place for 2 to 4 hours, until the layers of the rolls are puffy.
PREHEAT: Preheat the oven to 375°F. If your oven can't fit both sheets, put one in the refrigerator while the other is baking.
MAKE THE EGG WASH: In a small bowl, beat the egg and water together until combined. Brush over the dough, getting between the layers but avoiding getting any egg wash on the foil.
BAKE: Place one of the baking sheets in the preheated oven. Bake for 20 to 25 minutes or until golden brown. Remove to a cooling rack and swap in the other sheet of rolls.
COOL: Cool for 10 to 15 minutes, then carefully remove the foil cones.
SECOND BAKE: Bake for 5 more minutes without the foil. This will increase the cones' durability when you're filling them.
Final Cool: Transfer the warm rolls to a wire rack to allow them to cool down for 30 to 60 minutes before filling.