In the bowl of a stand mixer, whisk together the flour, sugar, salt, and yeast. In a bowl, add the yeast to the water; add this to the dry ingredients in bowl. Add the oil, eggs, and egg yolk. Using the dough hook attachment, mix at low speed until the dough forms.
On medium-low, and continue kneading until the dough becomes soft, usually 6 minutes. The dough should not be sticky; add additional flour or water as needed.
Turn the dough out onto a lightly floured surface. Pull the edges into the center to shape into a ball. Place dough in a large, lightly-oiled bowl, rolling it around to coat it with oil, leaving the dough smooth side up. Cover tightly with plastic wrap, then with a clean, dry towel, and let stand at room temperature for 1 hour. I turn the oven to 175 degrees for a couple of minutes, then turn it off, place the dough in the oven with the door ajar, then close the door when most of the heat has escaped.
Keeping the dough in the bowl, fold one end up, then the other end down, then crosswise, turning over to keep the smooth side on top. Cover again with plastic wrap, then towel and let rise another hour (it should just about double in size). You can use the above technique again to help it rise.
Turn the dough out onto a lightly floured surface, and gently deflate with your fist. Divide it into 3 equal pieces and form each into a long rope on countertop. Cover loosely with plastic wrap and let rest on the counter for 10 minutes.
Line a baking sheet with parchment paper. Roll each rope into a strand, all of the same length. Braid three strands together. Transfer the loaf to the prepared pan.
Beat the remaining egg white with 1 tablespoon water to make an egg wash. Brush the loaf with the egg wash. Cover loosely with lightly oiled plastic wrap, and let sit at room temperature until nearly doubled in size, about 70 minutes. 30 minutes before baking, preheat the oven to 350°, placing the rack in the middle position of the oven.
Gently brush the loaf again with the egg wash, taking care not to deflate. Bake at 350º F for 20 minutes. Rotate the pan 180º and continue baking for 20 to 45 minutes, until the bread is a rich golden brown and hollow sounding when you tap on it. Cool on a wire rack to for about an hour before serving.