01 - In a large mixing bowl, combine the warm water, sugar, and instant dry yeast. Stir gently and let the mixture rest for 5 minutes until it becomes foamy and bubbly on the surface.
02 - Add the all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough begins to come together. Turn the dough 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 kneaded dough to a greased bowl, turning once to coat all sides. Cover tightly with plastic wrap or a clean kitchen towel and set in a warm, draft-free spot for 1 hour until the dough has doubled in bulk.
04 - Preheat the 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 12×16-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 filled dough tightly into a log. Using a sharp knife, slice the log in half lengthwise down the center. Arrange the two halves cut-side up and braid them together by crossing one strip over the other, keeping the filling exposed on top. Pinch the ends together firmly to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the surface generously with the beaten egg to promote a rich golden crust during baking.
08 - Bake on the center rack for 28 to 32 minutes until the top is deeply golden and the cheese is visibly bubbling between the braids.
09 - Remove from the oven and immediately brush the hot loaf with melted butter. Sprinkle with additional Parmesan and fresh chopped basil if desired. Allow the bread to cool for 10 minutes before slicing into portions and serving.