1. For the pizza dough: Process flour, sugar, yeast, and salt in a food processor until combined, about 3 seconds. With the processor running, slowly add cold water and oil and process until dough forms a sticky ball that clears sides of the bowl, 30 to 60 seconds.
2. Transfer dough to lightly oiled counter and knead until smooth, about 1 minute. Shape dough into a tight ball and place in a greased bowl. Cover bowl with plastic wrap and let dough rise at room temperature until almost doubled in size, 2 hours.
For good results, let the dough rest at room temperature until doubled in volume, about 2 hours. For better results, refrigerate the dough overnight. For best results, refrigerate the dough for at least 3 and up to 5 days. If refrigerating the dough, remove it from the refrigerator and let sit, covered, at room temperature before proceeding to Step 4.
3. For the sauce: Heat the oil in a small saucepan over medium heat. Add the garlic, red pepper flakes and Italian seasoning and cook until the garlic becomes fragrant, about 1 minute. Stir the tomato paste and cook for 1 minute longer. Add the tomatoes, sugar and salt and cook for 5 minutes. Let cool before putting on the pizza.
(Sauce can be refrigerated for up to 2 days.)
4. One hour before baking and prepping, heat your pizza oven to 500 degrees using an oven thermometer. Transfer dough to lightly floured counter, divide in half, and gently shape each half into balls. Return 1 dough ball to the bowl and cover with plastic.
5. Coat the remaining dough ball lightly with flour and gently flatten into an 8-inch disk using your fingertips. Using a rolling pin, roll dough into a 12‑inch circle, dusting dough lightly with flour as needed. Prick the entire surface of the pizza with a fork, transfer to a piece of parchment paper and refrigerate for at least 30 minutes and up to 2 hours.
6. Sprinkle the pizza peel with cornmeal. Transfer dough to prepared pizza peel and carefully stretch to return to a 12-inch circle. Using a ladle, spread scant 1/2 cup sauce in a thin layer over the surface of dough, leaving 1/8-inch border around the edge; sprinkle three tablespoons of the parmesan over the sauce.
7. Pinch 6 ounces sausage into approximate dime-size pieces and evenly distribute over sauce. Sprinkle 1 1/2 cups mozzarella evenly over sausage to the edge of the pie. Distribute half of the giardiniera over the cheese to the edge of the pie. Sprinkle 1/4 teaspoon oregano over top.
7. Carefully slide pizza onto baking stone and bake, until cheese is well browned and edges of pizza are crisp and dark, 8 to 12 minutes, turning the pizza every minute or so for even browning.
8. Slide pizza peel underneath the pizza and remove pizza from the oven and onto a cutting board; let cool for 5 minutes. Repeat with remaining dough, sauce, sausage, mozzarella, and oregano. Cut pizzas into 2- to 3-inch squares and serve.
