Mix bread flour, sugar, instant yeast, salt, and grated parmesan cheese together in a large bowl or the bowl of your stand mixer
Add in milk, water, a beaten egg, and olive oil.
Stir together into a shaggy dough.
Knead on your stand mixer until it pulls away from the sides of the bowl and you can shape it into a smooth ball that is tacky but not sticky. If kneading by hand, the dough should have the same consistency.
Place in a lightly oiled bowl, cover, and place in a warm environment until it has doubled in size, about 20 to 30 minutes
Roll to a rectangle about 22" x 8", stretching the ends of the dough as needed to square it off.
Spread sun-dried tomato pesto evenly over the surface, leaving about a 1 inch border all around.
Hand tear basil leaves all over the surface.
Grate 2 ounces of Parmesan cheese over top of everything.
Starting from the long edge, use a gentle pushing motion to tuck and roll the dough, ensuring a tight spiral that hold all of the fillings.
Pinch the seam and both ends shut tightly.
Place the log of dough diagonally across a large parchment paper covered sheet pan, seam side down. Starting 1 inch from the near end, cut the log lengthwise down the center, about ½ inch deep, stopping about 1 inch from the other end.
Gently twist the two halves around each other to form an "S" shape or a "figure 8" keeping the filling side facing up. Tuck both ends of the dough under the center of the "S" and pinch them together to seal.
Cover and let rise another 20 minutes while you preheat the oven to 350°F. Brush exposed surfaces of dough with an egg wash consisting of one egg whisked with 1 tablespoon of water.
Bake at 350°F for 35-40 minutes. Bake until the internal temperature reaches 200°F. Cool on a wire rack before slicing.