While chicken is cooking, prepare the garlic mayo (you can shortcut these steps by using store-bought mayo, and adding minced garlic to it).
Smash and peel the garlic cloves, and give them a rough chop. Add to a food processor.
Half the lemon, and squeeze the juice into a food processor.
Add the room temperature egg to the food processor, and mix until smooth. (if your egg isn't room temp, put it in a glass with warm water and let it sit for 5-10 minutes).
With the food processor on, very slowly pour in the olive oil until combined.
Add salt, pepper and extra lemon juice to taste, and mix to combine. Set aside 1/2 cup for the chicken salad, and store any extra in the fridge for up to 2 weeks.