Mince the parsley, scallions, and dill in a food processor fitted with a steel blade.
Add the anchovies, garlic, capers, asiago cheese, and lemon juice. Process until smooth.
With the machine running, add the oil in a thin, steady stream.
Season to taste with salt and pepper.
Remove to a serving bowl and stir in the eggs.
(Makes about 1 ½ cups)