01 - In a large mixing bowl, combine the warm water, sugar, and instant dry yeast. Let the mixture rest for 5 minutes until it becomes foamy and active.
02 - Add the all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough forms. Turn the dough onto a floured surface and knead for 7 to 10 minutes until it is smooth and elastic.
03 - Place the kneaded dough in a greased bowl, cover with a damp towel or plastic wrap, and let it rise in a warm spot for 1 hour until doubled in size.
04 - Preheat the oven to 375°F. Line a baking sheet with parchment paper and set aside.
05 - Punch down the risen dough and roll it out on a floured surface into a 12 by 16 inch rectangle. Spread the basil pesto evenly across the surface, leaving a small border around the edges. Sprinkle the shredded mozzarella and grated parmesan uniformly over the pesto layer.
06 - Starting from the long edge, roll the dough tightly into a log. Using a sharp knife, slice the log in half lengthwise. With the cut sides facing upward, braid the two halves together by crossing them over each other, then pinch the ends together to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the surface evenly with the beaten egg wash.
08 - Bake for 28 to 32 minutes until the crust is golden brown and the cheese is bubbling.
09 - Remove the bread from the oven and immediately brush with melted butter. Sprinkle with extra parmesan and chopped fresh basil if desired. Allow to cool for 10 minutes before slicing and serving.