Make herb oil.
In a small bowl, stir together the olive oil, basil, oregano and salt. Set aside until ready to use.
Shape the pizza dough.
Place pizza stone on a grill or in the oven and heat to 500ºF (260ºC). Divide the dough in half. Lightly flour a work surface with semolina flour, working one half at a time, press out each half of the dough into a 10” (25cm) circle.
Work one pizza at a time. Use a pizza peel to slide the dough onto the pizza stone. Rub half the olive oil over the dough and cook until the crust is a pale brown and bubbly, about 6 minutes.
Spread pizza sauce evenly over the pizza, leaving about 1” (2.5cm) around the edges for the crust. Top with basil leaves and ¼ cup Parmesan cheese. Continue to cook until the crust is golden brown and the Parmesan has melted, about an additional 6 minutes.
Transfer the pizza to a cooling rack so the crust remains crisp. Top with an additional ¼ cup Parmesan cheese followed by pieces of torn burrata. Garnish with a drizzle of the herb oil then cut into pieces and serve. Repeat with the remaining pizza dough and toppings.