In the bowl of a stand mixer fitted with the paddle attachment, beat together the eggs, brown sugar, and orange blossom water until the sugar dissolves, 2 to 3 minutes. Add both flours, the yeast, and salt and mix until combined. Switch to the dough hook and, with the mixer on medium speed, add the butter 1 tablespoon at a time, waiting until each addition is almost completely mixed in before adding the next, then continue mixing until all the butter is fully incorporated and the dough is elastic and sticky, 10 to 15 minutes. You should be able to pull on the dough without it immediately breaking.

Butter two large bowls. Divide the dough in half, shape each piece into a ball, and transfer to the bowls. Cover with plastic wrap or kitchen towels and set aside in a warm (72°F to 77°F/22°C to 25°C), draft-free place until almost doubled in size, 1 hour to 1 hour and 15 minutes.

Generously butter two 9-by-5-inch (23-by-13-cm) loaf pans. Gently transfer one piece of risen dough to a clean work surface. Gently stretch the two opposite sides of the dough and fold over into the center, then repeat with the top and bottom to form a round; be careful not to tear or deflate the dough too much. Gently roll it into a 9-inch (23-cm)-long log and nestle it, seam side down, in one of the pans. Repeat with the second piece of dough. Cover loosely with plastic wrap and set aside to proof in a warm, draft-free place until the dough rises to the tops of the pans, 1 hour and 15 minutes to 2 hours.

Position a rack in the lower third and preheat the oven to 350°F (180°C).

Using a pastry brush, brush each loaf with the egg wash and then, using a pair of scissors, snip each top 5 or 6 times lengthwise down the middle. Let rest for 5 minutes, then place the pans in the oven- and bake until golden, 45 minutes to 1 hour 30 minutes. Check the loaves after 30 to 40 minutes and cover loosely with aluminum foil if the tops are browning too quickly.

When the loaves are done, immediately remove them from the pans and let cool completely on a wire rack before slicing.

Wrapped in linen cloths or in paper bags, the brioche will keep for 3 to 5 days. For longer storage, wrap well in plastic wrap and freeze for up to 3 months.