Preheat oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3-4 minutes until softened.
Add minced garlic and cook for 1 minute until fragrant.
Add ground turkey to the skillet and cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes.
Add diced red bell pepper, zucchini, and sliced mushrooms to the skillet. Cook for 5 minutes, stirring occasionally, until vegetables are tender.
Stir in drained diced tomatoes, dried Italian seasoning, salt, and black pepper. Cook for 2 minutes to combine flavors.
Remove skillet from heat and stir in cooked brown rice until evenly mixed.
Transfer the mixture to a 9×13 inch baking dish.
Sprinkle shredded mozzarella cheese evenly over the top, then sprinkle grated Parmesan cheese.
Cover the baking dish with aluminum foil and refrigerate if making ahead, up to 24 hours.
When ready to bake, preheat oven to 375°F (190°C) if not already heated.
Bake covered for 25 minutes, then remove foil and bake an additional 15 minutes until cheese is melted and bubbly.
Let casserole rest for 5 minutes before serving.