In a stand mixer, whisk bread flour, whole-wheat flour and yeast together. Add 1¼cups warm water, 2tbsp melted butter and honey. Using the paddle attachment on the mixer, mix on low speed for 1 minute. Increase speed to medium and mix for 2 minutes.
Scrape down the bowl and paddle with a greased spatula. Continue to mix for 2 minutes longer. Remove bowl and paddle from mixer. Scrape down the bowl and paddle, leave the paddle in the bowl. Cover with a plastic wrap and let rise in a warm place until doubled in volume, about 20 minutes.
Preheat the oven to 375°F. Lightly grease a 8½”x4½” loaf pan with cooking spray.
Dissolve salt in 2tbsp warm water.
When the batter has doubled, attach bowl and paddle to mixer. Add the salt-water mixture and mix on low speed until water is mostly incorporated, about 40 seconds.
Increase speed to medium and mix until thoroughly combined, about 1 minute, scraping down paddle if necessary.
Transfer batter to prepared pan and smooth surface with greased rubber spatula.
Cover and leave in warm place until batter reaches ½” below edge of pan, 15~20 minutes. Uncover and let rise until center of batter is level with edge of pan, 5~10 minutes longer.
Gently brush top of risen loaf with egg mixture (or soy milk). Bake until deep brown and loaf registers 208~210°F, 40~45 minutes. Carefully invert the bread onto a wire rack and brush the top and sides with the remaining 1tbsp melted butter. Cool completely before slicing.