In the bowl of an electric mixer fitted with the dough hook, combine flour, sugar, salt, and yeast.
Stir together.
Add egg yolks, vegetable oil, and water.
Mix on low speed for 3 minutes.
Increase speed to medium.
Knead for an additional 5 minutes. (The dough should be smooth and shiny. You can use the windowpane test to check for proper gluten development.)
Scrape dough into a large bowl.
Cover with plastic wrap. (If using a proofing box with a water tray, there is no need to cover with plastic wrap.)
Let rise at warm room temperature (78℉-80℉) for 2 hours.
Turn dough out onto an unfloured surface.
Degas by patting down gently.
Divide into thirds.
Form each third into a rectangle.
Cover with a piece of plastic wrap or a damp tea towel.
Allow dough to rest for 15 minutes.
While dough is resting, make egg wash by whisking egg yolk and water together until thoroughly combined and smooth. Set aside.
Working with one rectangle at a time, turn dough horizontally.
Fold the top third down and the bottom third up.
Turn over so seam is on bottom.
Roll into a log that is 12 inches long.
Place on a large sheet of parchment paper.
Repeat with 2 remaining pieces of dough.
Braid, making sure to seal and tuck ends under. (See step-by-step photos in post.)
Brush with egg wash.
Cover with plastic wrap that has been sprayed with cooking spray.
Allow to rise in a warm place for 1 hour, until noticeably risen.
While dough is proofing, heat oven to 350℉.
Brush dough with egg wash again.
Bake for 18-22 minutes, or until done (190℉).