Make the dough. Add 250g flour and 1 (¼ oz) packet of instant yeast to the bowl of a stand mixer fitted with hook attachment. Mix on low speed until blended. Add ⅔ cup milk, 1 large egg, 2 Tbsp neutral oil, 1 tsp Kosher salt, and ½ tsp granulated sugar. Mix on low speed for 30 minutes (yes, really!). Check after a few minutes to make sure all ingredients have been incorporated. If not, stop the mixer, scrape down the sides, then continue mixing. Meanwhile, make the pesto.
Make the basil pesto. Place 1 garlic clove and ¼ cup pine nuts in the bowl of a food processor fitted with blade attachment. Pulse until very finely chopped, then add 2 cups of basil. Pulse until well blended. With the motor running, slowly drizzle in ½ cup olive oil until a smooth sauce forms. Stir in ½ cup grated Parmesan, then set aside.
Fold the dough. Turn dough out onto a lightly floured surface, pushing down on the dough to release any bubbles. Gently pull the top of the dough down onto the bottom. Next, pull the right side of the dough over towards the left side. Pull the bottom of the dough up over to the top. Finally, stretch the left side of the dough over towards the right.
Rest the dough. Turn the dough over so the seams are underneath. Smooth out the dough into a round ball, then transfer to a bowl sprayed with nonstick oil. Cover with plastic wrap and let sit for 20 minutes.
Spread pesto. Turn the dough out onto a lightly floured surface. Use a rolling pin to roll dough out into a roughly 16" x 10" rectangle. Spread the pesto into a thin layer across the dough, leaving only a ½” border along the bottom. (Add ½ cup shredded mozzarella on top, if using.)
Roll the dough. Dip your finger in water, then run it along the bottom exposed part of the dough. Roll the rectangle from top to bottom lengthwise.
Shape the bread, then let it rest. Leaving about 1-2″ intact at the top, take a sharp knife and cut the dough in half lengthwise. Turn the dough cut side up, then twist to form a braided pattern. Place rolled dough into a loaf pan (10½" x 5½" x 3″) lined with parchment paper. If the dough is too large, tuck the ends underneath. Cover with plastic wrap and let sit for 2–2½ hours.
Bake, then cool. When ready, preheat an oven to 375°F (190°C). Brush the loaf with egg wash. Bake for 45-50 minutes, or until dough is browned and slightly crispy on the outside and fully cooked on the inside. Transfer to a wire rack to cool for 1 hour before slicing.