Place a pizza stone or baking sheet in oven. Preheat oven to 500°F. (Do not remove pizza stone while oven preheats.)
Combine vinegar, 1 cup water, sugar, and salt in a saucepan. Bring to a simmer over medium-high; cook, stirring occasionally, until sugar and salt dissolve. Stir in carrots, cucumber, radishes, and jalapeño. Remove from heat; let stand 10 minutes. Drain and set aside.
Roll dough into a 13-inch circle on a large piece of parchment paper; pierce well with a fork. Brush soy sauce and 1 tablespoon sesame oil over dough. Place dough (on paper) on preheated stone. Bake at 500°F for 6 minutes.
Sprinkle mozzarella evenly over dough. Bake at 500°F for 5 minutes.
Combine mayonnaise, Sriracha, and remaining 1 1/2 teaspoons sesame oil.
Spread pickled vegetables evenly over pizza. Drizzle mayonnaise mixture over top. Sprinkle with cilantro. Cut into 8 slices.