To the bowl of a stand mixer fitted with the hook attachment, add 1 cup flour, yeast, and milk. Using a spatula, mix until well-combined, then cover with plastic wrap and let sit for 45 minutes. Once the sponge has formed some air pockets, add in eggs, remaining 3 cups flour, sugar, and salt.
Mix on medium speed until well combined, then gradually increase to medium-high speed and continue mixing until dough pulls away from the sides of the bowl and becomes shiny and elastic, scraping down bowl every 4 to 5 minutes, for about 10 to 13 minutes.
With the mixer running, add in butter gradually, 1 tablespoon at a time, letting each tablespoon fully incorporate into the dough before adding the next, for 13 to 15 minutes. Continue mixing on medium-high speed for 5 to 7 minutes until the dough passes the windowpane test. Cover bowl with plastic wrap and let rest about 1 hour or doubled in size.
To bake on the Next Day: Once the dough has doubled in size, punch down to deflate it completely, then cover again with a plastic wrap. Refrigerate overnight until you are ready to bake the next day. Follow instructions in the next step, letting the dough proof until doubled before baking, longer if needed, up to 2 hours 30 minutes.
To bake on the Same Day: Once the dough has doubled in size, turn out onto a floured surface and punch down dough. Divide into half using a bench scraper. Cut each half into six equal pieces. Flatten each piece into a rectangle, then fold short ends in towards each other as if folding a letter. Flatten again and tightly roll into a log starting with the short end. Repeat with all pieces.
Grease 8x5-inch loaf pans with butter. Place 6 pieces of the dough seam-side down in one straight row into each prepared pan. Cover with plastic wrap.
Preheat the oven to 375 degrees. Let the dough proof until puffy and doubled in size, 1 to 1½ hours. In a small bowl, whisk together the remaining egg and water. Brush some egg wash on top of the loaf and sprinkle lightly with salt.
Bake until deeply golden on top and the center of the loaf registers between 190° and 205°, about 30 minutes.
Let the loaves cool for 5 minutes then turn them out onto a cooling rack. Let cool completely.