01 - In a large mixing bowl, combine the warm water, sugar, and yeast. Let the mixture stand for 5 minutes until it becomes foamy and fragrant, confirming the yeast is active.
02 - Add the flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough comes together. Turn out onto a lightly floured surface and knead for 7 to 10 minutes until the dough is smooth, elastic, and springs back when pressed.
03 - Place the dough in a lightly oiled bowl, turning once to coat. Cover tightly with plastic wrap or a damp kitchen towel. Allow to rise in a warm, draft-free spot for 1 hour, or 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 transfer to a floured surface. Roll out into a 12 by 16-inch rectangle. Spread the basil pesto evenly across the surface, leaving a small border around the edges. Scatter the mozzarella and Parmesan cheeses 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 in half lengthwise, keeping the two halves connected at one end. With the cut sides facing up, twist the two halves around each other to form a braid, pinching the ends together to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the entire surface generously with the beaten egg wash. Bake for 28 to 32 minutes until the crust is deep golden brown and the cheese is melted and bubbling.
08 - Remove from the oven and immediately brush the hot crust with melted butter. Garnish with additional Parmesan and chopped fresh basil if desired. Allow the loaf to rest for 10 minutes before slicing into portions.