Place a rack in top-most position of oven; preheat to 475 degrees. Place dough on a work surface; drizzle with 1 tablespoon oil, turning to coat. Stretch to a 10" round.
Heat 1 tablespoon oil in a large cast-iron skillet over medium. Add mushrooms and cook until soft and beginning to brown. Transfer mushrooms to a small bowl.
Remove skillet from heat and carefully lay dough inside. Carefully stretch dough to edges of pan. Season with sea salt, then top with mozzarella, mushrooms and garlic slices. Drizzle with another tablespoon oil. Peek underneath the crust—the bottom should be golden brown and crisp from residual heat in the skillet. If not, set over medium-low and cook until bottom crust is golden brown, about 3 minutes.
Transfer skillet to oven and bake pizza on top rack until crust is golden brown around the edges and cheese is browned in spots and bubbling all over, 10–14 minutes. Let cool 5 minutes, then top with red pepper flakes and mizuna. Sprinkle with more salt, a drizzle of olive oil and serve.