In the bowl of a standing mixer, combine warm water, honey, and yeast. Stir until all grains dissolve. Let proof for 10 to 15 minutes until yeast is puffy.
Add the sourdough starter discard to the yeast mixture and stir until well incorporated. Let proof for another 10 to 15 minutes.
In a separate container, whisk together the rye and wheat flour (and vital wheat gluten if needed) and salt.
Add the flours to the yeast and use the dough hook on low to combine.
While the dough hook is running, add the egg and melted butter. Mix for 5 to 10 minutes until the dough pulls away from and cleans the sides of the standing mixer bowl.
Turn the dough out onto a lightly greased surface and shape into a round.
Transfer the round to a lightly greased container with lid. Cover and let rise until doubled in bulk and the dough is light and puffy, about 2 1/2 hours.
Turn the dough out onto a lightly greased surface. Shape into a sandwich loaf, and transfer to a lightly greased 9-inch by 5-inch baking pan. Cover with plastic wrap and let rise until dough breaches the lip of the pan, about 1 hour.
Preheat oven to 350° Fahrenheit (176° Celsius).
Bake loaf for 20 minutes. Tent with aluminum foil to prevent browning, and bake for an additional 25 minutes. Internal temperature should read between 195° to 200 °Fahrenheit (90 to 93° Celsius), bottom of the loaf should sound hollow when thumped, and crust will have a nice light golden color.
Turn out loaf onto a wire cooling rack and allow it to cool completely before slicing and serving.
Handle the dough gently during shaping! The whole wheat and rye flours need a delicate touch, or you'll tear the dough and end up with holes in your crust.