1. In heavy-bottomed skillet over medium-high heat, add 1 tablespoon vegetable oil. Add onion all at once; do not stir. Cook onion until edges begin to brown, reduce heat to medium-low, stirring occasionally, 5 to 8 minutes or until caramelized.
2. Place potatoes in medium saucepan; add water just to cover. Bring to boil, reduce heat and simmer 1 minute; pour off water. When cool enough to handle, grate potatoes with largest grate on box grater. Toss lightly with salt.
3. Heat 2 tablespoons oil and butter in 10-inch skillet over medium heat. Add potatoes; fry until crispy. Turn. Set aside.
4. Heat oven to 400°F. Meanwhile, in 8-inch nonstick ovenproof skillet, add remaining 1 tablespoon oil; heat over medium heat. Add ham and caramelized onions; cook until hot. Pour in eggs all at once; sprinkle with cheese. Season with salt and pepper.
5. Bake 8 to 10 minutes or until center is set and top is slightly brown. Serve frittata with hash browns.