01 - In a large mixing bowl, combine the warm water, sugar, and yeast. Let the mixture rest for 5 minutes until it becomes foamy and fragrant, indicating active yeast.
02 - Add the flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough begins to come together. Turn out onto a floured work surface and knead for 7–10 minutes until the dough is smooth, supple, and springs back when pressed.
03 - Place the dough in a lightly greased bowl, cover with a damp towel or plastic wrap, and let rise in a warm spot for 1 hour or until doubled in volume.
04 - Preheat the oven to 375°F. Line a baking sheet with parchment paper and set aside.
05 - Punch down the risen dough and transfer to a floured surface. Roll out into a 12×16-inch rectangle. Spread the basil pesto evenly across the surface, leaving a small border around the edges. Sprinkle the mozzarella and 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, slice the log lengthwise down the center to create two halves. With the cut sides facing up, intertwine the two halves by crossing them over each other, forming a braid. Pinch 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 for a golden, glossy finish.
08 - Bake on the center rack for 28–32 minutes until the crust is deeply golden and the cheese is melted and bubbling through the braid.
09 - Remove from the oven and immediately brush the hot loaf with melted butter. Sprinkle with additional Parmesan and fresh basil if desired. Allow to cool for 10 minutes before slicing into portions and serving.