Preheat the oven to 375°F (190°C).
In a large saucepan, heat the olive oil over medium heat. Add the onion and garlic, sautéing until the onion becomes translucent, about 3-4 minutes.
Add the broccoli, bell peppers, zucchini, and corn to the pan. Sauté for an additional 5 minutes until the vegetables are slightly tender. Remove from heat.
In a large mixing bowl, combine the cooked rice, sautéed vegetables, eggs, milk, dried oregano, salt, and black pepper. Mix well until all ingredients are combined.
Fold in 1 cup of shredded cheddar cheese and ½ cup of Parmesan cheese into the mixture.
Transfer the mixture into a greased 9x13 inch baking dish, spreading it evenly.
Sprinkle the remaining 1 cup of cheddar cheese and ½ cup of Parmesan cheese over the top.
Bake in the preheated oven for 30-35 minutes, until the cheese is bubbly and golden brown.
Once cooked, remove the dish from the oven and let it cool for 5 minutes before serving.