Place a pizza stone or inverted baking sheet in the lower third of the oven and preheat to 450 degrees F. Mix the flour, baking powder and 1/2 teaspoon salt in a medium bowl. Stir in 1/3 cup water and 2 tablespoons olive oil until a shaggy dough forms. Turn out onto a lightly floured surface and knead into a smooth ball, 1 to 2 minutes. Wrap in plastic wrap and set aside for 15 minutes at room temperature.
Meanwhile, toss the beans, giardiniera and brine, basil, parsley and 1 tablespoon olive oil in a large bowl. Season with salt and pepper; set aside. Slice the tomatoes, place on paper towels and sprinkle lightly with salt; set aside.
Roll out the dough into a 12-inch round on a lightly oiled piece of parchment paper. Slide the parchment with the dough onto an inverted baking sheet. Rub 1 tablespoon olive oil and the garlic all over the dough. Sprinkle the parmesan on top. Slide onto the hot stone and bake until browned and crisp, about 8 minutes.
Remove the crust from the oven and top with the mozzarella, tomatoes and basil leaves. Drizzle with the remaining 1 tablespoon olive oil and season with salt and pepper. Serve with the bean salad.