In a large bowl, combine the flour, yeast, sugar, and salt. In a smaller bowl, beat together the water, eggs, egg yolks, and oil. Carefully combine the mixtures in a bowl or in your stand mixer fitted with the dough hook.
Knead the dough, either on a lightly floured work surface with your hands or with the dough hook. Initially you will have a shaggy mess with a lot of dry flour but this will eventually give way to a dough that is very elastic (this ought to take 10 to 20 minutes). It may be necessary to add a touch more water if not all the flour can easily be incorporated—start by adding a tablespoon water, work the dough a little, and then if necessary add another tablespoon.
When the dough is smooth and elastic, form it into a smooth ball, place it in a large bowl slicked with oil, and cover with plastic wrap. Let it rise at room temperature until it has doubled in volume, about 1 1/2 hours.
Divide the dough into 6 equal portions. Using the palms of your hands, shape each portion into evenly shaped sausages about 12 inches (30 centimeters) long.
On a rimmed baking sheet that’s lightly floured or lined with parchment paper, braid the dough. (For spectacularly helpful instructions on how to braid challah dough, check out this advice from the folks at King Arthur Flour on how to braid 6 strands of dough—without needing 6 hands.) Sprinkle the braid lightly with flour, then cover loosely with plastic wrap. Let it rest at room temperature until it has doubled in volume, about 1 hour. Preheat the oven to 350°F (180°C).
In a small bowl, whisk together the egg white and sugar and lightly brush it over the braid. Bake the challah bread for 25 to 35, minutes until the bread has risen considerably and is golden brown. Let cool slightly before slicing.