In the bowl of a stand mixer fitted with the dough hook attachment, add the all-purpose flour, water (except for 80 grams or 1/3 cup), sugar, fresh yeast, and 2 of the eggs.
Mix on low speed until a rough dough forms.
Slowly add the remaining water and switch to medium speed while doing so.
The dough will come apart but let it keep mixing until it comes back together with the water, about 3 minutes.
Once the water is incorporated, add the butter 50 grams (3 1/2 tablespoons) at a time and switch to high speed. Make sure the butter is incorporated before adding more. With the last bit of butter, add the salt (and the vanilla). Let the mixer continue on high speed for 4 to 5 minutes until you get a smooth, creamy dough.
Transfer to a clean, buttered bowl, cover and let ferment for 4 hours.
After the initial fermentation, dump the dough onto a floured work surface and divide it into twenty-four 80-gram pieces (about the size of clementines).
Shape into balls and then use the palms of your hands to roll into tight logs, about 3 inches (8 centimeters) long.
Place the rolls on parchment-lined sheet pans, 3 by 4, cover with a plastic bag or a large damp cloth, and proof for 2 hours.
Combine the remaining egg with a pinch of salt and whisk rigorously until homogenous.
Brush each pebete with the egg wash and bake for 25 minutes at 375˚F (190˚C). They should have a nice, dark brown shine to them when they are done.