01 - In a large mixing bowl, combine warm water, sugar, and yeast. Let stand for 5 minutes until the mixture becomes foamy and fragrant.
02 - Add flour, salt, and olive oil to the yeast mixture. Stir until a sticky dough comes together, then turn out onto a floured surface.
03 - Knead the dough for 7 to 10 minutes until smooth and elastic. Place in a greased bowl, cover with a damp towel, and let rise for 1 hour until doubled in bulk.
04 - Preheat oven to 375°F. Line a baking sheet with parchment paper.
05 - Punch down the risen dough and roll it out to a 12 by 16-inch rectangle. Spread the basil pesto evenly across the surface, leaving a small border around the edges. Sprinkle mozzarella and parmesan cheese in an even layer over the pesto.
06 - Starting from a 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 into a braid, pinching the ends together to seal.
07 - Carefully transfer the braided loaf to the prepared baking sheet. Brush the entire surface with beaten egg for a golden finish.
08 - Bake for 28 to 32 minutes until the crust is deeply golden and the cheese is bubbling through the layers.
09 - Remove from the oven and immediately brush with melted butter. Top with additional parmesan and fresh basil if desired. Allow to cool for 10 minutes before slicing.