In the bowl of an electric mixer fitted with the dough hook, combine the flour, yeast, sugar, honey, salt, vegetable oil, egg yolks, and 1 cup water. Mix on low speed until the dough is smooth and stiff with a slight sheen, 8 to 10 minutes.
Turn the dough onto a lightly floured work surface, and knead to make sure any loose bits are incorporated. Fold in the following manner: Fold the bottom third of the dough up, the top third down, and the right and left sides over, tapping the dough and flip it over, seam side down. Place the dough in a lightly oiled bowl, cover with oiled plastic wrap, and let rise in a warm place until doubled in bulk, about 1 hour. Brush an unrimmed baking sheet with vegetable oil or line with parchment paper. Set aside.
Return the dough to a lightly floured work surface, and divide into three equal pieces. Roll each piece into an 18-inch log, and place the logs parallel to one another; pinch the ends together at the top. Weave the three strands into a tight braid, tugging gently as you go. Press the ends together to seal. Place loaf on the prepared sheet. Loosely cover with oiled plastic wrap, and let rise until doubled in bulk, about 1 hour.
Meanwhile, preheat the oven to 375°F. Lightly beat the remaining whole egg and brush gently but thoroughly over loaf, making sure to cover any seams and crevices. Bake until the crust is dark brown, 50 to 60 minutes. The bread should reach an internal temperature of 190°F on an instant-read thermometer, and should have a hollow sound when tapped on the bottom. Transfer the bread to a wire rack to cool completely before slicing. Bread can be wrapped in plastic and kept at room temperature for up to 3 days.