Feed your starter. Feed your sourdough starter the night before by combining the sourdough starter, all-purpose flour, and water. Store at room temperature for about 12 hours or until it has doubled in size.
Make the roux. Add the ¼ cup all-purpose flour and ½ cup milk to a saucepan. Place over medium heat, using a whisk, stirring constantly, and allow the mixture to thicken and become a paste. This process takes about 3 minutes. Transfer to the bowl of your stand mixer and cool to about 80-90℉.
Make milk mixture for dough. To a saucepan, add the milk, salt, and sugar. Place the milk mixture over medium-low heat and bring it to 80-90℉. You want the milk to be slightly warm but not hot.
Combine with roux. Transfer the milk mixture to the bowl of your stand mixer with the roux already added. Using the paddle attachment, stir to combine.
Add sourdough starter. Next, add the active sourdough starter. Stir to combine.
Add bread flour. Add 2 cups of the bread flour to the mixture and stir to combine. Slowly add the rest of the bread flour.
Add egg. The dough will be very wet and sticky. Add the egg and stir to combine fully.
Slowly add all-purpose flour. Switch the attachment to the dough hook and slowly add the all-purpose flour to the dough by adding ½ cup at a time, incorporating the flour with each addition.
Add butter. Once the flour is fully incorporated, add the softened butter. Stir to combine.
Knead. Set your stand mixer to medium speed and knead for 3 minutes. You will notice the dough taking shape. It should still be sticky but holding its shape.
Cover and rest. Transfer the dough to a greased bowl, cover, and allow to sit for 30 minutes.
Stretch and folds. After 30 minutes, remove the cover from the bowl, wet your hands, and grab the dough from one side. Stretch the dough up and fold it over, turn the bowl slightly, and repeat this process 2 or 3 more times until the dough has been released from the bowl. Cover and allow the dough to sit for 30 minutes. Repeat this process 1-3 more times. Your dough should be soft and stretch without breaking.
Bulk rise. Cover and allow the dough to rise for 6-12 hours or until the dough has doubled in size.
Prepare pans. Grease 2 loaf pans, place a piece of parchment paper inside, and set them aside.
Divide dough. Turn the dough out onto a clean, dry surface. Using a bench scraper, divide the dough in half. Take each half and divide it into 3 balls.
Shape dough balls. Lightly flatten each ball into a rectangle, take one end and gently roll it up about halfway, fold in the sides, then finish rolling it up. Transfer the rolled dough to the prepared loaf pan. Repeat this process until all 3 balls have been rolled.
Second rise. Cover and allow to rise for 2-4 hours or until the loaves have doubled in size.
Preheat oven. Preheat the oven to 375℉.
Top with egg wash. Make the egg wash in a bowl by adding an egg and water. Stir to combine. Using a pastry brush, brush on the egg wash.
Bake. Bake at 375℉ for 35-40 minutes or until the tops are golden brown.
Cool. Remove from the oven, place them on a cooling rack, and allow to cool for 20 minutes before serving.