Remove the reserved rectangle of polenta from the refrigerator and let stand at room temperature.
Preheat broiler (or toaster oven broiler).
Heat 2 tsp. oil in a large skillet over medium-high heat. Add mushrooms; cook, stirring often, until softened and lightly browned, 4 to 6 minutes. Stir in garlic, thyme, and salt.
Broil the polenta until warmed and lightly toasted, 3 to 5 minutes. Top with the mushroom mixture, mozzarella, and Parmesan. Broil for 2 to 3 minutes more.
Meanwhile, whisk the remaining 1 tsp. oil and lemon juice in a medium bowl. Add arugula and black beans; toss to coat. Add avocado and gently toss. Divide the salad between 2 plates.
Sprinkle the pizza with parsley, cut into 6 pieces and serve with the salad.
Baked Turkey Meatballs with Roasted Red Peppers & Polenta