Add the flour, salt, and sugar to the bowl of a stand mixer fitted with the paddle attachment. With the machine running, add the starter, eggs, and milk. Mix on low until a sticky dough forms. Approximately 2 to 3 minutes. Cover the dough and allow to rest for 15 to 30 minutes in the mixing bowl.
Change the paddle attachment to a dough hook. With the mixer on low speed, add the butter one cube at a time, waiting a few seconds between each addition until you see the butter cube mix in part of the way. It won’t all disappear and that’s okay.
Once all the butter is in, increase the speed to medium and knead the dough 5 to 7 minutes more. The dough should be shiny and smooth but still slightly loose in the bowl and will not form a ball.
Transfer the dough to a lightly buttered or greased bowl. Cover with a damp towel or oiled plastic wrap and allow the dough to rise 5 to 8 hours depending on the temperature of the room. 70 degrees is best. Once risen, cover with oiled plastic wrap if you have been using a towel previously and transfer to the refrigerator to chill overnight.
The next morning, lightly grease a 9 x 5 x 3 inch loaf pan with butter or oil. Transfer the cold dough to your countertop and divide the dough into either 4 pieces at approximately 265g each or into six pieces that are about 175 grams.
Working with one piece at a time gather the ends of the dough on the bottom then gently roll into a ball by forming your hand into a “claw” shape over the round of dough and moving your hand in a tight circle. Stagger the dough into the loaf pan seam side down if you are using four balls. If using six, place them in two rows of three balls each. The dough will fit quite tightly.
Cover the dough with a damp towel and let the dough rise until risen 1 inch above the rim of the loaf pan; approximately 1 1/2 to 2 hours or more. Because the dough is cold it could take significantly longer. Just watch the dough and not the clock. The dough should rise to approximately 1 inch above the rim of the loaf pan.
Preheat the oven to 400°F. Mix the egg and a splash of water together in a bowl and use a brush to generously coat the surface of the risen dough.
Bake on the center rack 40 to 45 minutes. Tent with foil halfway through the bake time if the dough starts to get too dark. Cool in the pan for 10 minutes, and then transfer to a wire rack to cool completely before cutting into slices. The loaf will stay fresh for up to two days stored at room temperature.