3 tablespoons extra-virgin olive oil, plus more for the bowl
3 cups all-purpose flour, or 2 3/4 cups plus 1/4 cup whole-wheat flour, plus more for dusting
1 teaspoon fine salt
For the toppings:
3 cups sliced green cabbage
1 teaspoon pickling spices, tied securely in cheesecloth
1 large potato, peeled and thinly slicedFreshly ground pepper
2 cups shredded mozzarella cheese
3/4 cup shredded monterey jack cheese
1/2 cup freshly grated parmesan cheese
6 ounces sliced corned beef
Make the dough: Whisk 1 cup warm water (105 degrees) with the sugar in a bowl; scatter the yeast over the top and set aside until foamy, about 10 minutes. Stir in the olive oil.
Whisk the flour and salt in a large bowl. Make a well in the center and pour in the yeast mixture. Gradually stir the dry ingredients into the wet ingredients to make a rough, shaggy dough. Turn out onto a floured surface and knead until smooth and elastic, about 5 minutes. (Add more flour to prevent sticking, if necessary.) Form the dough into a ball; place in a large oiled bowl, turning to coat with oil. Cover the bowl with plastic wrap and set aside at room temperature until the dough has doubled in size, about 90 minutes.
Meanwhile, prepare the toppings: Heat 1 tablespoon olive oil in a skillet over medium-high heat. Add the cabbage, season with salt and cook until just soft, about 5 minutes. Add the pickling spices and just enough water to cover. Simmer over low heat, covered, until the cabbage is tender, about 20 minutes. Drain the cabbage and set aside (discard spices).
Place a pizza stone in the oven, if you have one, and preheat to 500 degrees. Toss the potato with 2 tablespoons olive oil and season with salt and pepper. Roast in a single layer on a baking sheet until golden, about 15 minutes.
Divide the dough into 2 equal pieces. Roll one into a 14-inch round (keep the remaining dough covered). Place the round on a floured pizza peel (if baking on a stone) or a large oiled pizza pan; drizzle with 2 tablespoons olive oil. Scatter half of each of the cheeses, corned beef, cabbage and potatoes on top. Season with salt and pepper. Carefully slip the pizza onto the hot stone, if using, or place the pan in the oven. Cook until golden and crispy, 10 to 15 minutes. Repeat with the remaining dough and toppings.