Preheat the oven to 425 degrees F (220 degrees C).
Spray 4 (6 ounce) ramekins with cooking spray. Place 4 ounces mashed potatoes into each ramekin. Using the back of a spoon, mold into nests, and brush with 1 lightly beaten egg.
Break 1 of the 4 remaining eggs into each ramekin. Drizzle each with 1/2 teaspoon milk, top with 1 pat of butter, and 1 tablespoon Cheddar cheese.
Cook until yolk is starting to set and cheese is melted, 15 to 20 minutes (cooking time will depend on how you want your eggs cooked). Remove from the oven, and let stand until residual heat finishes cooking the egg whites, 1 to 3 minutes.
Season with salt and pepper, garnish with green onions or chives, and serve immediately.
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