Step 1

To make the water roux, in a medium saucepan, combine the water, milk and flour, then whisk until lump-free.Set over medium and cook, whisking constantly, until the mixture thickens (a silicone spatula drawn through the mixture leaves a trail) and bubbles slowly, 2 to 4 minutes. Scrape into a medium bowl, press a sheet of plastic wrap directly against the surface and cool to room temperature.

Step 2

To make the dough, brush a large bowl with melted butter; reserve the remaining melted butter. Add two of the eggs to the cooled roux and whisk until well combined. Add the room-temperature milk and whisk until homogeneous and smooth.

Step 3

In the bowl of a stand mixer, whisk together the bread and rye flours, sugar, milk powder, yeast and salt. Attach the bowl and dough hook to the mixer and, with the machine running on low, slowly add the roux-egg mixture.With the mixer still running, add the softened butter 1 tablespoon at a time. Increase the speed to medium-low and knead until the dough is very strong and elastic, 10 to 12 minutes; it will stick to the sides of the bowl. Scrape the dough into the prepared bowl, then brush the surface with melted butter. Cover the bowl with plastic wrap and let rise in a warm spot until doubled in bulk, about 1½ hours. Meanwhile, coat 2 metal 8½-by-4½-inch loaf pans with melted butter.

Step 4

Lightly flour the counter. Gently punch down the dough, then turn it out onto the prepared counter. Using a chef’s knife or bench scraper, divide the dough into 4 equal portions, each about 355 grams (about 12½ ounces).Shape each portion into a smooth ball. Using your hands, pat one ball into a 7-by-4-inch rectangle, then fold the dough into thirds like a business letter. Pinch the seam to seal.Turn the dough seam side down and place on one side of one of the prepared loaf pans so the seam is perpendicular to the length of the pan. Shape a second portion of dough, then place it in the pan alongside the first portion, positioning it the same way; there should be just a small amount of space between the 2 pieces of dough. Cover the pan with a clean kitchen towel.

Step 5

Repeat the process with the remaining portions of dough, then place under the towel alongside the first pan. Let rise until the dough domes 1 to 1½ inches over the rim of the pan, about 1 hour. Meanwhile, heat the oven to 350°F with a rack in the middle position. In a small bowl, whisk the remaining egg until well combined; set aside.

Step 6

When the dough is properly risen, gently brush the tops with the beaten egg. Bake until the loaves are well risen and golden brown, 30 to 35 minutes. Cool in the pans on a wire rack for 15 minutes. Gently invert the bread out of the pans, stand them upright on the rack and cool for at least 1 hour before slicing.