Whisk instant yeast and one tablespoon brown sugar in lukewarm water until dissolved. This can be done easily by hand, however, it is easiest if you use the bowl of a stand mixer as the mixer is very helpful for later steps.
Add oil, continuing to whisk.
Add 4 eggs, salt, and remaining 1/2 cup sugar, alternating between eggs and continuing to whisk.
Using the dough hook on your kitchen aid, or a wooden spoon if you do not have a stand mixer, gradually add flour. Keep the mixer on low speed and add one cup at a time to reduce mess. If you are using vital gluten, add it at this time as well. Add enough flour so the dough is no longer sticky and does not easily take more flour (see photos and tips above).
Continue kneading for several minutes until smooth either with the dough hook or by hand.
Remove the dough ball from the bowl, grease the bowl, and return the dough. Cover with plastic wrap and let rise until doubled in size. The time this takes depends on the temperature and humidity in your room. It should, however, be at least an hour and a half or two.
Once the challah has doubled in size punch down and allow to rise for another forty-five minutes.
Divide challah dough and braid challah to the desired shape. Four and six-strand loaves are both pretty. If you are using stuffings such as chocolate chips flatten the strands after forming them, add the chocolate chips, and pinch the sides up (see photo above).
Preheat oven to 375.
Allow challah to rise for another hour.
Crack and scramble the final egg in a bowl and brush it on your loaves. If you are using a topping (such as everything but the bagel spice) sprinkle on now.
Bake in the oven for 30-35 minutes, depending on the loaf size and your desired doneness. At that point you can poke the challah to make sure it is baked inside.
Turn the oven to broil and allow the tops to brown for an additional 2-5 minutes. Monitor to make sure the tops do not burn.
Remove from oven and allow to cool.