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Spray a rubber spatula with cooking spray. Butter a large mixing bowl and set aside.
Spray a 8½x4½-inch loaf pan with cooking spray.
Fit a stand mixer with the paddle attachment. In the bowl of the mixer, whisk bread flour, whole-wheat flour and yeast.
Add the warm water, 3 tablespoons melted butter and honey.
Mix on low 1 minute.
Increase the speed to medium and mix 2 more minutes. Scrape down the dough from the bowl and the paddle with the prepared spatula and mix 2 more minutes.
Transfer the dough to the buttered bowl and rotate it so all sides of the dough are buttered. Cover the bowl with plastic wrap and set in a warm place to rise until doubled in size (20-30 minutes).
Place oven rack to lower-middle position and heat oven to 375°F.
Dissolve salt in 2 tablespoons warm water and set aside.
When dough has doubled in size, transfer the dough to the mixer bowl, and reattach the mixer bowl to the mixer stand. Add the salt-water mixture, and mix on low just until water is incorporated (about 45 seconds).
Increase speed to medium and mix 1 minute, scraping down the sides of the bowl and the paddle when necessary.
Transfer the batter to the prepared pan and smooth out the surface with the greased spatula.
Cover with damp light-weight, kitchen towel and place in a warm place until dough rises to ½-inch below edge of the pan (about 15 minutes).
Uncover and let rise arise another 5-10 minutes or until the center of the batter is even with the edge of the pan.
Carefully brush the top of the loaf (after is has risen) with the egg mixture.
Bake until golden brown and/or the internal temperature registers 208-210 degrees Fahrenheit on an instant-read thermometer, 35-45 minutes.
After removing bread from the oven, carefully invert it onto a wire cooling rack. Turn it upright and brush the top and sides with 1 tablespoon melted butter.
Cool completely before slicing.