For the pizza dough: Place the all-purpose flour, salt, sugar and yeast in the bowl of a food processor and pulse to combine. With the motor running, slowly add the olive oil and water; once added, continue mixing for 1 minute.
Transfer the dough to a work surface and knead until smooth.
Divide the dough into 3 even pieces and form into balls. Spray with cooking oil and then place in a large resealable plastic bag. Refrigerate overnight or up to 4 days for best flavor. (See Cook's Note.)
When ready to bake, transfer each dough ball to a lightly oiled bowl, cover and set aside until doubled in size, about 1 hour.
Make the pizza pies: Preheat a gas grill to 500 degrees F or get a charcoal grill as hot as you can. Lightly dust a work surface with all-purpose flour.
For each pizza: Gently flatten a piece of dough with the palm of your hand and then roll it into a 10- to 11-inch round, about 1/8 to 1/4 inch thick. Don't worry if it's not perfectly or even barely round; just make sure it's of uniform thickness (even if it's the shape of a beaver tail!).
Sprinkle a pizza peel with semolina flour and transfer the stretched dough to the peel. (If you don't have a peel, use the back of an upturned sheet pan instead.) Slide the dough directly onto the grill. Cook until golden grill marks appear, about 1 minute per side; use tongs to flip.
For the white pizza: In a small bowl, mix together the ricotta and fontina cheeses along with the thyme, rosemary, honey and some salt and pepper. Brush a grilled crust with olive oil. Spread with the cheese mixture in an even layer. Use the peel to return the pizza to the grill. Cover and cook for about 3 minutes, until the cheese is completely melted and the crust is fully browned. Remove from the grill and top with the prosciutto and arugula.
For the margherita pizza: Brush a grilled crust with olive oil and spread evenly with the marinara sauce. Top with the torn mozzarella and sprinkle with the Parmigiano-Reggiano. Use the peel to transfer the pizza to the grill. Cover and cook until the cheese is melted and the crust is golden brown. Remove from grill and top with torn basil.
The dough makes enough for three pizzas. You can either double the topping ingredients to make a third pizza or freeze the third dough ball for future use.