Preheat the oven to 350°F (175°C).
In a large skillet over medium heat, cook the ground beef until browned, breaking it apart with a spoon, about 5-7 minutes.
Add the diced onion and minced garlic to the skillet with the beef. Cook until the onion is translucent, about 3-4 minutes.
Stir in the frozen mixed vegetables and cook for an additional 3 minutes until they are slightly tender.
In a medium bowl, combine the condensed cream of mushroom soup, sour cream, milk, salt, black pepper, and dried Italian seasoning. Mix well.
Add the cooked rice to the skillet with the beef and vegetables. Stir to combine.
Pour the soup mixture into the skillet and stir until all ingredients are evenly coated.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle shredded cheddar and mozzarella cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 30 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.