In a large bowl, dissolve 4 1/2 teaspoons (2 packages) active dry yeast and 1 tablespoon of the brown sugar in 1 3/4 cups lukewarm water. Let sit 5 minutes, until foamy.
Whisk 1/2 cup olive oil into the yeast mixture. Add the remaining 1/2 cup brown sugar, 1 tablespoon kosher salt, and 4 of the eggs (reserve the 5th egg for the egg wash). Beat or whisk until combined.
Add the flours gradually: the 4–4 1/2 cups whole wheat pastry (or white whole wheat) flour and the 4 cups all-purpose flour. Stir a cup at a time until the dough begins to hold together. You may use a sturdy stand mixer with a dough hook for mixing and kneading, or mix in a large bowl and finish by hand.
Turn the dough onto a lightly floured surface and knead until smooth and elastic, about 5–10 minutes. Add small amounts of flour only as needed to prevent sticking.
Clean the mixing bowl, lightly oil it with additional olive oil, place the kneaded dough back in the bowl, and turn once so the top is lightly oiled. Cover the bowl with plastic wrap or a kitchen towel.
Let the dough rise in a warm place for 1 hour, until almost doubled in size. (If you prefer, you may refrigerate the dough overnight at this point; if refrigerating, allow it to come to room temperature before continuing.)
Punch down the dough, cover, and let it rise again in a warm place for 30 minutes. (You may also refrigerate overnight here; if you do, bring to room temperature and allow the dough to finish rising before shaping.)
To shape a 3-braid challah, divide the dough in half. Work with one half at a time: divide that half into 3 equal pieces, form each into a ball, then roll each ball into a strand about 12 inches long and 1 1/2 inches wide. Pinch the top ends together and braid by bringing the right outside strand over the center, then the left outside strand over the new center, repeating until you reach the ends. Pinch and tuck the ends to seal. Repeat with the other half of dough.
Place the braided loaves on a lightly oiled baking sheet. Beat the reserved 5th egg and brush the loaves lightly with some of the beaten egg (reserve a little egg wash for brushing again before baking, if desired). Cover the loaves and let them rise for about 1 hour, until puffy and nearly doubled. (Or place in the fridge to rise overnight.)
If baking immediately, preheat the oven to 375°F while the loaves finish their final rise. Just before baking, brush the loaves again with any remaining beaten egg.
Bake on the middle oven rack for 30 to 40 minutes, or until the loaves are golden brown. Remove from the oven and cool completely on a wire rack.
The challah can be frozen either unbaked or baked.