Put the bread flour, sugar, salt, and yeast in the bowl of a stand mixer fitted with a dough hook. Add the pâte fermentée, egg yolks, honey, water, and oil and mix on low speed until the dry ingredients are completely incorporated (i.e., you can’t see any flour) and the yeast has disappeared into the dough. Add a little extra water if this hasn’t happened in 3 minutes. Increase the speed to medium to medium-high and mix until the dough is smooth, pulls away from the sides of the bowl (and leaves the sides clean), has a bit of shine, and makes a slapping noise against the sides of the bowl, about 5 minutes. Do the windowpane test to check to see if the gluten is fully developed. The dough will look smooth and feel slightly tacky.

Put the bread flour, sugar, salt, and yeast in the bowl of a stand mixer fitted with a dough hook. Add the pâte fermentée, egg yolks, honey, water, and oil and mix on low speed until the dry ingredients are completely incorporated (i.e., you can’t see any flour) and the yeast has disappeared into the dough. Add a little extra water if this hasn’t happened in 3 minutes. Increase the speed to medium to medium-high and mix until the dough is smooth, pulls away from the sides of the bowl (and leaves the sides clean), has a bit of shine, and makes a slapping noise against the sides of the bowl, about 5 minutes. Do the windowpane test to check to see if the gluten is fully developed. The dough will look smooth and feel slightly tacky.

Tip the dough out onto a lightly floured surface. Flatten slightly and divide it into 4 equal pieces (each weighing about 7½ ounces/215 g). Working with one piece at a time (keep the rest covered with plastic), form a tight log roll. Then, with two hands, give the piece a few rolls on the floured surface so that it forms a thick rope about 18 inches/45 cm long. Repeat to make 4 ropes.

Leave 2 ropes loosely covered with plastic. Take the first 2 ropes and form a two-strand braid (see photographs). Form an “X” in front of you with one rope going from the upper right to the lower left on the bottom, and the other on top, going from the upper left to the lower right on top. Fold the top right arm of the X down over the center so it’s now facing down toward you. Fold the bottom left arm of the X up over the center so it’s now where the top right arm used to be. Do the same with the top left arm and the bottom right arm. Keep building your braid in this fashion until you have no dough left to cross. Turn the braid on its side so that what was the base is now one end of the loaf and squeeze the small end pieces of dough firmly together and tuck them under the braid. Set the braided loaf on a parchment-lined rimmed baking sheet and repeat the process with the remaining 2 ropes of dough to form a second loaf. Evenly space the loaves apart on the baking sheet.

Carefully brush the challahs with the beaten eggs, reserving whatever egg is left over for a second egg wash. Put the entire baking sheet in a large plastic bag or cover the challahs loosely with plastic wrap, and let them stand at room temperature until they have risen, are supple, and hold indentations when pressed lightly, about 1 hour.

Meanwhile, preheat the oven to 350°F/180°C.

Uncover the challahs and gently brush them again with the reserved egg. Bake the loaves until they’re mahogany colored and sound hollow when you tap on the bottom of the loaves, 45 minutes to 1 hour. Insert a thin knife in between the strands to make sure that the dough is firm—it should have the density of a well-baked cake.

Transfer the breads to a wire rack to cool completely, at least 1 hour. Store leftovers in a plastic bag at room temperature or freeze.