Combine the water and yeast in a small bowl. Let set for 10 minutes, then stir until the yeast is completely dissolved. Set aside.
Sift together the flours, sugar, and salt into the bowl of a mixer fitted with the dough hook. Add the eggs and beat for 1 minute at low speed, scraping down the sides with a rubber spatula as needed. Slowly add the dissolved yeast and continue beating at low speed for 5 minutes. Stop the machine, scrape any dough off the dough hook, and beat for another 5 minutes. Add the butter cubes, about one quarter of them at a time, beating for about 1 minute after each addition. Once all the butter has been added, beat for 10 minutes more, until the dough is smooth and silky.
Transfer the dough (it will be very wet and soft) in a large floured mixing bowl and cover with plastic wrap. Set aside in a warm place until doubled in size, about 3 hours.
Turn the dough out onto a generously floured work surface and gently work the air bubbles out by folding the dough several times while lightly pressing down on it. Return the dough to the bowl, cover with plastic wrap, and refrigerate overnight.
Generously butter two 8 ½-by-4 ½-by-3-inch loaf pans. Turn the dough out onto a floured work surface. With floured hands, divide the dough in half and shape it into two rectangles to fit the loaf pans. Place the dough in the pans and let the dough rise uncovered in a warm place until it is about ½ inch above the top of the pans, about 3 hours.
Preheat the oven to 350°F. Bake the brioche until it is well browned on top and sounds hollow when tapped on the bottom, 35 to 40 minutes. Remove from the oven and immediately turn the brioche out onto a wire rack.
If using immediately, let the breads cool for 10 minutes, then slice. If serving within a few hours or up to 2 days, promptly wrap the hot bread in aluminum foil and set aside at room temperature until ready to use. To freeze, wrap the hot bread in foil and promptly freeze; when ready to use, reheat (without thawing, and still wrapped in foil) in a 250°F oven until heated through, 20 to 25 minutes. The bread can be kept frozen for up to 2 months.