Drain potato, reserving 1 cup cooking liquid. Mash the potato in a small bowl and let cool slightly.
Cream butter, sugar, salt, and egg in an electric mixer. Add potato and mix well.
Dissolve yeast in the 1/2 cup warm water. Add to potato mixture.
Change beaters to dough hooks, if you have them. Set mixer to 2. Add 2 cups whole wheat flour alternately with the 1 cup cooking liquid, mixing well after each addition. Gradually add in the remaining flour until the dough starts to clean the side of the bowl. When that happens, let mixer go an additional 2 minutes.
If your dough looks a little shiny, great! You’re ready. If not, you may need to knead it for a few additional minutes. (I have a Sunbeam Mixer, which is not quite as hardcore as a KitchenAid and sometimes I think it’s going to explode before the bread is really ready.)
Place the dough in a greased bowl and put somewhere warm. Cover with a damp towel. Let rise for an hour or until dough is doubled.
Punch dough down; divide in half. Shape into loaves. (I sort of push it into a rectangle, roll it up, and then tuck in the ends.) Place in two greased loaf pans. Cover with the towel again and let rise another 30-45 minutes or until doubled again.
Brush dough with melted butter. Bake at 350 for 25 minutes or until golden brown. Remove from pans and let cool on a wire rack.
And then, in your best Julia Child voice, say, “Bon Appetit!”