Place chicken breasts into a saucepan with enough water to cover them. Bring to a boil, then cover, and simmer for about 20 minutes, or until cooked through. Drain, and set aside to cool.

While the chicken is poaching, whisk together egg yolks and garlic until light colored. Whisk in olive oil in a thin stream until fully incorporated. Flavor with lemon juice and salt to taste. This is the aioli sauce.

Shred or cut chicken into small pieces, and place in a salad bowl. Toss with the olives, sun-dried tomatoes, and thyme. Stir in 3/4 cup of the aioli until everything is evenly coated. Taste and adjust seasoning if desired. Refrigerate, or serve immediately. Make a pleased groaning sound as you eat it.