In a huge bowl, dissolve yeast and sugar in 2 cups warm water and let it sit for about 15–20 minutes until thick and frothy. (I use a 13-quart bowl.)
Add eggs, honey, oil, salt, three cups of water remaining, and half the flour. Mix until a loose batter forms. Add the rest of the flour in a couple of cups at a time. You may not need all 18 cups of flour, so go slowly toward the end. Alternatively, you may need slightly more. The dough should be soft but not sticky. Once the dough has enough flour, knead it for a few minutes. I do this in the bowl. (You can make this recipe by hand or with a mixer.)
Cover the dough with a wet towel or plastic wrap and place it warmly to rise for about 1½ hours. The dough should double in size.
Punch the dough down and let it rest for 10 minutes. Divide into six equal pieces.
To braid the loaves, click on the link in the notes below to view the braiding instructions.
Preheat the oven to 375° F.
Place loaves on lightly greased pans and let rise for another 40 minutes.
Beat the egg with the honey and vanilla and gently brush over the loaves.
Bake at 375° F for approximately 45 minutes. The loaves should be golden on top and firm on the bottom.