1. Using an egg topper, carefully cut a lid from each egg. Empty eggs into a bowl, reserving 4 shells. Bring a small pan of water to a boil. Rinse reserved egg shells; carefully lower them into the boiling water. Simmer 3 minutes. Remove shells, and let them dry upside down on a towel.
2. Combine butter and eggs in a medium nonstick saucepan over medium-high heat, stirring eggs constantly with a flexible heatproof spatula or wooden spoon until they start to thicken (about 3 minutes). The eggs will start to coagulate suddenly, often around the sides or base of the pan. Lift pan off the heat, and stir rapidly so that any cooked egg dissolves back into the liquid part, stirring for about 30 seconds. Return pan to heat; cook about 3 minutes or until eggs thicken slightly, stirring constantly. Remove pan from heat, and stir 30 seconds. Continue placing the eggs over heat for about 3-minute intervals and removing for 30 seconds, stirring constantly until desired degree of doneness (repeat procedure about 5 times total). Sprinkle eggs evenly with salt and black pepper.
3. Divide egg mixture evenly, and carefully spoon into 4 prepared egg shells. Top each serving with 1/2 teaspoon crème fraîche and 1/2 teaspoon caviar. Set the eggs in egg cups, and serve at once.