01 - In a large mixing bowl, combine the warm water, sugar, and instant dry yeast. Stir gently and let stand for 5 minutes until the mixture becomes foamy and fragrant.
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 lightly floured surface.
03 - Knead the dough for 7 to 10 minutes until it is smooth, supple, and springs back when pressed. Place in a lightly oiled bowl, cover with a damp towel or plastic wrap, and let rise in a warm spot for 1 hour until doubled in volume.
04 - Preheat the oven to 375°F. Line a large 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 up tightly into a log. Using a sharp knife, slice the log lengthwise down the center into two halves. Arrange the two halves side by side with the cut sides facing up, then braid them together by crossing one strip over the other, pinching the ends firmly to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the entire surface generously with the beaten egg wash to promote a deep golden crust.
08 - Bake on the center rack for 28 to 32 minutes until the crust is deeply golden and the cheese is melted and bubbling. Rotate the pan halfway through for even browning.
09 - Remove from the oven and immediately brush the hot crust with melted butter. Garnish with additional Parmesan and fresh chopped basil if desired. Allow the loaf to cool for 10 minutes before slicing into portions.