Egg-enriched, yeasty dough is layered with sweet butter to make brioche feuilletee, which is French for flaky.
Dough: Combine milk and yeast in a small liquid-measuring cup. Set aside until yeast is dissolved, about 5 minutes.
In the bowl of an electric mixer fitted with the dough hook attachment, combine flours, salt, and sugar. Add yeast mixture and 4 eggs; mix on medium speed until dough is smooth, about 6 minutes. Add room-temperature butter, 1 tablespoon at a time, mixing well after each addition. Continue mixing until dough is smooth and shiny, about 3 minutes more. Turn dough out onto a lightly floured surface and knead gently into a ball. Wrap tightly in plastic and refrigerate 1 hour.
Butter Package: Arrange cold butter in a single layer on a piece of plastic wrap and loosely wrap. On a sturdy work surface, use a rolling pin to beat until butter is slightly softened, then roll out to a 5-by-8-inch rectangle, adjusting plastic as necessary. Refrigerate until chilled, about 20 minutes.
On a lightly floured surface, roll out dough to a 7-by-18-inch rectangle. Place chilled butter package on bottom half of dough, leaving about a 1-inch border on three of the sides. Fold top half of dough over butter to cover, then pinch edges together to seal. Tuck overhanging dough edges under to form a neat package. Roll out dough once more to a 7-by-18-inch rectangle. Fold top third of dough down towards center, then fold bottom third on top (like a business letter). This completes a single turn. Rotate dough package 90 degrees and repeat rolling and folding process. Wrap well in plastic and refrigerate about 45 minutes. Perform two additional single turns on dough. Wrap well in plastic and refrigerate at least 1 hour or up to overnight.
Brush two standard 9-by-5-inch loaf pans with room-temperature butter. Roll out dough to a 14-by-14-inch square. Lightly brush top 3 inches of dough with water. Starting at opposite long edge, roll dough into a tight spiral, pressing to seal end. Cut log into 6 equal pieces and space evenly in prepared pans, cut-sides up. Lightly brush sides of dough pieces that face each other with water. Loosely cover with plastic and let rise in a warm spot until dough is light and full of air and has expanded to nearly fill pans, 1 to 1 1/2 hours.
Preheat oven to 350 degrees. Bake until loaves are puffed and lightly golden, about 35 minutes. Brush with beaten egg and continue baking until deep golden brown and an instant read thermometer inserted into center reads 205 degrees, 5 to 10 minutes more. Let cool in pans 5 minutes, then unmold onto a wire rack and let cool completely before serving.