Make The Dough. Add the flour, salt, and sugar to the bowl of a stand mixer fitted with the paddle attachment. Mix briefly to combine. With the machine running, gradually add the starter, eggs, and warm milk. Mix on low speed until a sticky, shaggy dough forms and all of the flour is fully absorbed about 2 to 3 minutes. Scrape down the sides of the bowl as needed. Cover and rest the dough for 15 to 30 minutes. Meanwhile, replenish your starter with fresh flour and water, store at room temperature or in fridge.
Add the butter. Fit stand mixer with dough hook attachment. On low speed, add the butter one cube at a time, waiting for 10 to 20 seconds before the next addition. Increase to medium speed and knead the dough until butter is fully incorporated about 5 to 7 minutes more. When ready the dough will look shiny and smooth but will not come together into a ball. It will also feel warm to the touch. Scrape down the the sides of the bowl as needed.
Bulk Rise. With floured hands, transfer dough to a new lightly buttered bowl. Cover the bowl with a damp towel and find a warm spot for the dough to rise. This will take about 5 to 8 hours (at 70oF, 21oC). Once fully risen, cover the dough with lightly oiled plastic and transfer to the fridge. Chill overnight.
Shape.In the morning, lightly coat a 9x5 inch loaf pan with butter. Remove the cold dough onto a lightly floured surface. Dough will be very firm. To shape into a loaf, pat the dough into a rectangle and roll into a log. Place in loaf pan, seam side down. Or divide the dough into 4 pieces, about 265 g each. Working with one piece at a time, gather the ends, flip the dough over, and gently roll into a ball. Stagger the dough, seam side down, into your loaf pan. The dough will fit snugly.
Second Rise. Cover the dough with a damp towel and let rise until puffy, about 1 1/2 to 2 hours. The dough is ready when it looks puffy, and has risen about 1 inch about the rim of the loaf pan.
Preheat. Preheat your oven to 400oF (200oc). Combine the egg with water and brush the top of the dough in the egg wash.
Bake.Bake the pan on the center rack for 40-45 minutes. If the loaf starts to brown too quickly, loosely tent with foil. The loaf will be rich, golden brown and shiny when finished. Cool in the pan for 10 minutes, then remove to a wire rack to cool completely before slicing.