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 activated.
02 - Add the all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough comes together, then turn out onto a floured surface and knead for 7 to 10 minutes until the dough is smooth and elastic.
03 - Place the kneaded dough into a greased bowl, cover with a damp towel or plastic wrap, and let it rise in a warm spot for 1 hour or 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 in an even layer over the pesto.
06 - Starting from one long edge, roll the dough tightly into a log. Using a sharp knife, cut the log lengthwise down the middle to create two halves. With the cut sides facing up, twist and braid the two halves together, pinching the ends firmly to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the entire surface with the beaten egg wash. Bake for 28 to 32 minutes until the crust is deep golden and the cheese is bubbling.
08 - Remove the loaf from the oven and immediately brush with melted butter. Garnish with extra parmesan and chopped fresh basil if desired. Allow the bread to cool for 10 minutes before slicing and serving.