Mushroom Flatbreads

1 1/2 pounds fresh mushrooms (such as oyster, beech, and/or cremini), cut into small pieces (about 7 cups)
6 tablespoons olive oil, divided
1 teaspoon kosher salt
1/2 teaspoon black pepper, plus more to taste
1 large garlic clove, grated
1/2 cup heavy cream, plus more as needed
2 tablespoons white miso
1 (16 1/2- x 4 1/2-inch) flatbread (such as Trader Joe’s Pizza Crust)
2 ounces low-moisture part-skim mozzarella cheese, shredded (about 1/2 cup)
1 1/4 ounces Parmesan cheese, grated (about 1/3 cup)
1 1/2 cups loosely packed baby arugula

Preheat oven to 450°F. Toss together mushrooms, 1/4 cup oil, salt, and pepper on a large rimmed baking sheet. Roast in preheated oven, stirring occasionally, until browned, 25 to 35 minutes. Remove from oven. (Do not turn oven off.)
Stir garlic into roasted mushrooms; set aside. Whisk together cream and miso in a small bowl until well combined. Set aside.
Brush both sides of flatbread with remaining 2 tablespoons oil; place on a parchment paper–lined large baking sheet. Bake at 450°F until toasted, about 5 minutes. Remove from oven. Spread flatbread with half of the miso-cream sauce. Top with half of the mushroom mixture. Sprinkle with mozzarella and Parmesan; top with remaining mushroom mixture. Return to oven, and bake at 450°F until bread is crisp and cheese melts, 6 to 8 minutes.
Thin remaining miso-cream sauce with additional cream as needed to reach a drizzling consistency. Toss together arugula and 1 teaspoon miso-cream sauce in a medium bowl.
Scatter arugula mixture over flatbread. Sprinkle with additional pepper to taste, if desired. Cut into 8 pieces, and serve alongside remaining miso-cream sauce.