Cook the carrots and the peas separately in the microwave for 2 to 3 minutes on high. Carrots should be soft, peas should be dark green. Chill thoroughly in the refrigerator.
Mix the mayonnaise, lime juice, and garlic in a mixing bowl. Adjust the amount of mayonnaise to get the right consistency.
In a medium mixing bowl, toss the carrots, peas, celery, onion, and cooked chicken together. Add the mayonnaise mixture a little at a time until you get the consistency you like.
Add salt and pepper to taste, and fold in the chopped eggs. Serve well chilled on a bed of lettuce.