Heat oil in medium oven-proof pan. Add corn and shallots; cook 3 minutes. Stir in 0.25 tsp salt, 0.25 tsp pepper, and all paprika. Remove corn mixture to bowl.
Whisk together eggs, milk, and remaining salt and pepper. Stir in corn mixture
Preheat broiler to high.
Melt butter to same pan. Add egg mixture, cook 1 minute without stirring. Gently slide pan back and forth to keep eggs from sticking. Cover, reduce heat to low, and cook 4 minutes until eggs are set and golden on bottom.
Sprinkle cheese evenly over top. Brown a few minutes until the cheese browns. Sprinkle with parsley.