To begin, combine the flour, yeast, and 1 1/3 tsp salt in your mixer's bowl. Fit your mixer with the dough hook.
In a dish or measuring cup, combine the 1/2 c water, sugar, and oil. Add 2 eggs plus the egg yolk. Whisk them together well.
Turn the mixer on low speed and very slowly pour the liquid mixture into the bowl. After about 2 minutes, scrape the bowl with a rubber spatula to make sure everything is combined.
Turn the mixer on low again and let it run for 10 minutes.
Lightly sprinkle your work surface with some flour and turn the dough out and gently knead it to form a ball.
Place the dough seam side down into a lightly oiled bowl. Cover the bowl with plastic wrap. Place the bowl in a warm location to rise for 1 1/2 - 2 hours, or until it has doubled in size.
Turn the dough out onto a lightly floured surface. Divide the dough into 3 or 4 equal portions. Roll each portion into a long rope, at least 14-16 inches.
You will form the loaf on the baking sheet or stone that is lined with parchment paper.
Braid the dough together, pinching the ends together. Tuck the ends underneath. Cover the loaf with plastic wrap and let it rise for an hour or 2, until it is doubled in size again.
Heat your oven to 375 degrees.
In a small bowl, whisk the remaining egg, tablespoon of water, and pinch of salt together.
Brush the egg wash all over the loaf.
Bake for 20-25 minutes, or until the loaf is beautifully bronzed.
Transfer the loaf to a rack and allow it to cool fully before cutting, about 2-3 hours.