Part 1
Dissolve the yeast in ½ cup of the warm water and a bit of the sugar in a large bowl. When it froths and bubbles, it is ready to use. Mix in the rest of the Part 1 ingredients, 2 cups AP flour, 3 cups whole what flour, 2/3 cup oil, 5 eggs, 1 1/2 tablespoons salt, 1/3 cup honey, 3/4 cup brown sugar and the rest of the water. Stir until they form a batter.
Cover the bowl and let it rise for 30-45 minutes.
Part 2
Punch the dough down and then add the ingredients listed in part 2; 5 cups whole wheat flour, 5 cups AP flour, 1 cup bran, and 1 cup warm water.
Knead the entire batter until you have a workable and smooth dough. If it is sticky, smear it with a bit more oil and then knead a bit more until it you reach the desired consistency.
Separate challah with a bracha (for instructions click here).
Let this dough rise another hour, covered in plastic. Then shape as desired.
Brush the risen challahs with a mixture of 1 whole egg + 1 egg yolk + 1 T. sugar.
Bake in a preheated oven at 350° F until golden brown top and bottom, about 40 minutes.