Combine flour with 150 grams water, salt and starter in the bowl of a stand mixer. Mix mixture until dough comes together. Add an additional 39 grams water to the dough and knead for 4-5 minutes or until a shaggy dough forms. The dough will not be smooth but it should be strong and hold its shape.Transfer the dough to a large bowl, cover and let proof for 30 minutes. Stretch and fold the dough 3-4 times and let sit for another 30 minutes. Repeat the stretch and fold method. Continue to follow this process for a total of 4 folds. After the last fold, let rest for 30 minutes. At this point, the dough should be fairly smooth and strong.
Shape the dough into a tight ball and transfer to a lightly oiled bowl. Cover with plastic wrap and let rest in the refrigerator overnight.
Remove the dough from the refrigerator and divide into two even balls. Shape each ball into a tight ball. Place the balls onto a tray and cover with plastic wrap. Let rest for 1 hour.
Preheat oven to 550 degrees F and warm the pizza stone for 1 hour.
Lightly dust a clean work counter and gently transfer one of the pizza balls onto the counter. Using your hands, press down on the dough, keeping your hands flat with the dough. Try not to press out any gas at the edge of the dough as this will create the air pockets. Shape and press into a flat disc. Top with half of the parmesan cheese, and bocconcini.
Switch the oven to broil on high. Carefully slide the pizza onto the pizza stone and broil for 1 minute. Turn off the broiler and turn on the oven to 550 degrees F. Continue to bake for 2-3 minutes or until lightly golden. Remove from oven. Top with chives, freshly grated black pepper and a drizzle of olive oil. Repeat with the remaining dough. Serve immediately.