1. Heat a medium ovenproof skillet over medium-high heat. Add oil to pan; swirl to coat. Add corn and shallots to pan; cook 3 minutes, stirring frequently. Stir in 1/4 teaspoon salt, 1/4 teaspoon pepper, and paprika, if desired. Place corn mixture in a bowl; cool slightly. Wipe skillet clean with paper towels.
2. Place remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, milk, and eggs in a bowl; stir with a whisk until well combined. Add corn mixture to egg mixture, stirring to combine.
3. Preheat broiler to high.
4. Return pan to medium heat. Add butter to pan; swirl until butter melts. Add egg mixture to pan. 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 or until eggs are set and golden on the bottom. Sprinkle cheese evenly over eggs. Broil 1 minute or until eggs are just set and cheese browns. Sprinkle evenly with parsley.