Corned-Beef-and-Cabbage Pizza

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For the dough:

2 teaspoons sugar

1 package active dry yeast

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:

5 tablespoons extra-virgin olive oil, plus more for the pan

3 cups sliced green cabbage

Kosher salt

1 teaspoon pickling spices, tied securely in cheesecloth

1 large potato, peeled and thinly sliced

Freshly 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