01 - In a large mixing bowl, combine the warm water, sugar, and instant dry yeast. Let the mixture stand for 5 minutes until it becomes foamy and activates.
02 - Add the all-purpose flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough begins to come together.
03 - Turn the dough out onto a floured surface and knead for 7 to 10 minutes until smooth and elastic. Place the dough in a greased bowl, cover with a clean towel or plastic wrap, and let rise for 1 hour until doubled in size.
04 - Preheat the oven to 375°F. Line a baking sheet with parchment paper and set aside.
05 - Punch down the risen dough and roll it out on a floured surface into a 12 x 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 uniformly over the pesto layer.
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 upward, twist and braid the two halves together, pinching the ends firmly to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the entire surface generously with the beaten egg for a golden finish.
08 - Bake in the preheated oven for 28 to 32 minutes until the crust is deep golden and the cheese is bubbling and melted.
09 - Remove the loaf from the oven and immediately brush with melted butter. Garnish with additional parmesan and chopped fresh basil if desired. Allow the bread to cool for 10 minutes before slicing and serving.