Gently place the eggs into a large saucepan and cover with cold water. Add the vinegar and place over high heat. Bring to a boil and set a timer for 5 minutes. Remove from the heat after 5 minutes and let sit an additional 2 minutes. Drain the eggs and run under cold water. Immediately peel the eggs* and slice in half the long way. Remove the yolks to a mixing bowl and set the whites to the side. Using a fork, mash the yolks well. Add the mayonnaise, anchovy paste, mustard and white pepper and mix well. Spoon the mixture into a pastry bag fitted with a #6 round tip (1/2-inch diameter). Alternatively, use a sealable plastic bag with the tip cut off.
Set the egg whites out in a row. Into half, spoon 1/4 teaspoon of the black caviar. Into the other half spoon 1/2 teaspoon of the red caviar. Pipe the filling on top of the caviar so that each resembles a full yolk. Garnish the black caviar stuffed eggs with a small amount of the red caviar and vice versa. Garnish each with a small sprig of dill and serve on a bed of mixed greens.
*We've found that the combination of adding vinegar to the water and peeling them while still hot is the best method for removing the shell easily.