Sprinkle the yeast over the water in a small bowl, add a pinch of your sugar. Stir to dissolve the yeast and let stand about 5 minutes. Whisk together 4 cups of the flour, rest of the sugar, and salt in a large bowl.
Make a divot in the center of the flour mixture and add the eggs, egg yolk, and oil. Use a fork to gently combine the egg and oil, bringing in a little bit of flour from the sides. Pour the yeast mixture in the center of the flour with the egg and oil.
Mix the yeast, eggs, and flour with a spoon until you form a dry dough that is difficult to mix. Knead the dough for 5-8 minutes. If the dough seems very sticky, add flour a teaspoon at a time until it feels tacky, but no longer like gum.
The dough has finished kneading when it is soft, smooth, and holds a ball-shape. Place the dough in a well-oiled bowl, and flip the dough once to coat with the oil. Cover with a kitchen towel, and let the dough rise until doubled in bulk, 1.5 to 2 hours.
Separate the dough into three or six equal pieces, depending on the type of braid you’d like to do. Roll each piece of dough into a long rope roughly 1-inch thick and 16 inches long. If the ropes shrink as you try to roll them, let them rest for 5 minutes to relax the gluten and then try again.
Gather the ropes and squeeze together at the very top. Braiding the 3 ropes together like braiding hair or yarn and squeeze the ends together until the whole loaf is braided.
Line a baking sheet with parchment and transfer the braided bread to the prepared baking sheet. Sprinkle the loaf with a little flour and drape it with a clean dishcloth. Let it rise until puffed and pillowy, about an hour.
About 20 minutes before baking, heat the oven to 350 F. When ready to bake, whisk the reserved egg white with a tablespoon of water and brush it all over the challah. Be sure to get in the cracks and down the sides of the loaf. Bake for 30 to 35 minutes. Allow cooling before slicing.