In a large bowl, combine the yeast, sugar, olive oil and water and let stand until foamy, about 5 minutes.
Add the salt and 3 cups of flour and stir until a really stiff dough forms.
Turn the dough onto a lightly floured work surface and knead until smooth, about 5 minutes.
Oil the bowl, return the dough to it and let stand, covered, until doubled in bulk, about 2 hours.
Punch down the dough. Roll it into 4 balls. Transfer the balls to a work surface, cover with a towel and let rest for 15 minutes.
In a blender, puree the canned tomatoes with the oregano; season with salt and pepper.
Light a grill.
Work with 1 ball of dough at a time and keep the rest covered: Roll the dough out onto a floured work surface to a 10-inch round, 1/4 inch thick.
Transfer the round to a large, oiled baking sheet and brush with oil.
Repeat with the remaining 3 balls of dough.
Set all of the toppings near the grill. Rub the grate with oil. Carefully drape two of the pizza dough rounds over the grill and cook over low heat until golden on the bottoms, about 3 minutes.
Working quickly, flip the pizza crusts and top each with one-fourth of the tomato sauce, parsley, mozzarella, sliced tomato, pancetta and Gorgonzola. Close the lid and grill over low heat until the pizzas are lightly charred on the bottoms, the cheese is bubbling and the pancetta is just beginning to brown, 4 minutes.
Transfer the pizza to a baking sheet. Repeat with the remaining dough and toppings. Return the pizzas to the grill, cover and rewarm until crisp, 1 minute.
In a bowl, toss the arugula with the remaining 1 teaspoon of oil and the balsamic vinegar and season with salt and pepper.
Scatter the figs and arugula over the pizzas and serve.