Press the pizza dough into a greased 9- to 10-inch cast-iron skillet and let rise until doubled in size, 20 to 30 minutes.
Meanwhile, preheat the oven to 425 degrees . In a large skillet, cook the bratwurst over medium heat until browned, about 5 minutes. Let cool before slicing into 3/4-inch-thick rounds.
In a large bowl, whisk the sugar into 1 tablespoon hot water. Whisk in the vinegar, 3 tablespoons olive oil, the celery seeds and 1/4 teaspoon each salt and pepper. Add the coleslaw mix and toss.
Press the bratwurst into the risen dough. Brush with the remaining 1 tablespoon olive oil; sprinkle the crust with the caraway seeds. Bake until puffed and browned, about 25 minutes. Let cool in the pan for 5 minutes. Slice, then drizzle with the mustard and serve with the slaw.