In the bowl of the mixer, equipped with a hook, combine the sifted flour with the yeast, then with the sugar and salt.
Beat the eggs and egg yolks in another bowl, then add the oil and water, and beat for a few seconds to obtain a homogeneous mixture.

Pour the liquid mixture into the center of the flour.
Run the mixer at medium speed and knead for 10 minutes or until the dough is soft and elastic to the touch.

Form the dough into a ball and put it in a lightly oiled bowl, making sure to turn it to coat the entire surface.
Cover with plastic wrap and let rise for 1 hour.

Do a fold by detaching the dough from the sides of the bowl, lifting it several times to degas it.
Reform the dough into a ball, cover the bowl with plastic wrap, and let rise for 1 hour, or until the dough has increased by at least 1 and a half times its volume.

Transfer the dough to a work surface and divide it according to the desired shapes: 6 and 5 pieces for a 6-strand braid and a 5-strand braid.
Shape each piece into a ball, cover with a towel, and let rest for 10 minutes to relax the gluten.

Braid.

Place the 2 loaves on a baking sheet lined with a silpat, leaving about 8 inches (20 cm) of space between them.
Brush the loaves with the beaten egg whites, mixed with a pinch of salt.
Spray lightly with vegetable oil and cover with plastic wrap, without tightening.

Let rise for 1 to 1 hour 15 minutes, or until the loaves have reached about 1 and a half times their volume.

Preheat the oven to 350 F (180 C).
Gently remove the plastic wrap, then brush the loaves again with the egg whites.
Sprinkle with sesame seeds or poppy seeds.

Bake for about 10 to 15 minutes, then rotate the baking sheet and continue baking for about 10 minutes, or until the loaves have a beautiful golden color.
Let cool on a rack.