01 - In a large mixing bowl, combine the warm water (about 100°F), sugar, and instant dry yeast. Stir gently and let the mixture sit for 5 minutes until it becomes foamy and fragrant, indicating the yeast is active.
02 - Add the all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough begins to form. Turn the dough out onto a lightly floured surface and knead firmly for 7 to 10 minutes, until the dough is smooth, elastic, and springs back when gently pressed.
03 - Place the kneaded dough into a greased bowl, turning once to coat all sides. Cover tightly with plastic wrap or a damp kitchen towel. Set in a warm, draft-free spot and let rise for 1 hour, or until the dough has doubled in bulk.
04 - Preheat your oven to 375°F. Line a large baking sheet with parchment paper and set aside.
05 - Punch down the risen dough to release air bubbles. On a floured surface, roll it 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 shredded mozzarella and grated Parmesan uniformly over the pesto layer.
06 - Starting from one long edge, roll the filled dough tightly into a log. Using a sharp knife, slice the log in half lengthwise so you have two long strips with the cut sides facing up. Carefully braid the two halves together by crossing one strip over the other, keeping the layered filling exposed on top. Pinch both ends firmly to seal.
07 - Gently transfer the braided loaf to the prepared baking sheet. Brush the entire surface generously with the beaten egg wash. Bake on the center rack for 28 to 32 minutes, until the crust is deeply golden and the cheese is bubbling through the braid.
08 - Remove the bread from the oven and immediately brush the hot crust with melted butter for a rich shine. Sprinkle with additional Parmesan and fresh chopped basil if desired. Allow the loaf to cool for at least 10 minutes before slicing into portions. Serve warm as an appetizer or alongside soup.