In a food processor, combine fresh herbs, walnuts, and garlic. Pulse to combine.
Slowly add olive oil while pulsing until combined well.
Add salt and pepper to taste.
For the Paleo Mayo:
In a food processor, combine 1/4 cup oil, egg, mustard powder, and salt and pulse to combine. Slowly stream in remaining oil until emulsified. Add lemon juice and pulse to combine.
For the chicken salad:
In a large bowl, combine chicken, paleo mayo (or yogurt), and pesto. Add salt and pepper to taste.