Preheat oven to 350°F (180°C).
In a large skillet, heat oil over medium heat. Add onion and cook, stirring occasionally, for 2 minutes. Add mushrooms, broccoli, tomatoes, and asparagus; cook, stirring occasionally, until veggies are tender-crisp, about 5 minutes. Remove from heat.
In a large bowl, whisk together eggs and milk until blended. Add onion powder and season with salt and pepper. Add vegetables, hash browns and cheese and stir to combine.
Pour egg mixture into a greased 9-inch (23 cm) square baking pan. Bake, uncovered until a knife inserted into the center comes out clean, 45 to 50 minutes.
Let stand for 10 minutes before cutting