01 - In a large mixing bowl, combine warm water (about 100°F), sugar, and instant dry yeast. Stir gently and let stand for 5 minutes until the mixture becomes foamy and bubbly.
02 - Add all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough forms. 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 - Transfer the dough to a lightly greased bowl, turning once to coat. Cover with a clean kitchen towel or plastic wrap. Let rise in a warm, draft-free spot for 1 hour or until doubled in size.
04 - Preheat 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 × 16-inch rectangle. Spread basil pesto evenly across the surface, leaving a small border around the edges. Sprinkle 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, slice the log in half lengthwise. With the cut sides facing up, intertwine the two halves by crossing them over each other, keeping the exposed filling facing upward. Pinch both ends together to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the surface evenly with beaten egg. Bake for 28 to 32 minutes until the crust is deep golden and the cheese is melted and bubbling.
08 - Remove from the oven and immediately brush the hot crust with melted butter. Sprinkle with additional Parmesan and chopped fresh basil if desired. Allow to cool for 10 minutes before slicing into portions. Serve warm.