How to Make It
Preheat oven to 500°F with rack in lowest position.
Combine pine nuts, miso, and zest in a food processor; process until finely chopped and combined. Add basil, pulse until combined, 4 to 5 times. With processor running, slowly add ¼ cup of the oil; process until well combined, about 30 seconds. Set aside.
Heat 2 tablespoons oil in a large skillet over medium-high. Add mushrooms and cook, stirring occasionally, until browned and tender, about 4 minutes. Add tomatoes, garlic, salt, and pepper. Cook, stirring occasionally, until tomatoes just begin to break down, about 4 minutes. Remove from heat.
Stretch dough into a long 12-x 9-inch oval and place on rimmed baking sheet. Brush both sides with oil and prick with a fork. Bake until lightly browned, about 6 minutes. Remove from oven and spread pesto over dough, leaving a ½-inch border. Top evenly with mushroom mixture. Return to oven and bake until crust is golden brown on top and bottom, 3 to 4 minutes. Top with crushed red pepper flakes.