The traditional aïoli, also known as the "butter" from the Provence has no basil in it and will always be made with raw garlic.
The recipe is very simple; eggyolks, lemonjuice, cloves of garlic, olive oil and s&p are put in a mortar and crushed and mixed into a creamy mixture. Of course you can make it in a machine. If it gets too thick, add a dash of hot water.
This sauce is mostly eaten with crudités (raw vegetables), hardboiled eggs, blanched veggies, cold cooked potatoes. I've never seen it being served with meat but many times with seafood. But, as always, you do what you like.
This small suggestion for your chickenbreasts may be too late, but just try and see!
When heating your frying pan with oil to cook your chicken, add also 5 whole cloves of garlic, still in their skin (!!) so they don't burn. Let the garlic in there all the time. Season and brown quickly your chicken on both sides on high fire. Reduce fire to low and cover the frying pan loosely with a sheet of aluminiumfoil. Let cook until done (8-10 minutes, depending). Remove the breasts from the pan and wrap in the alumiunfoil to rest.
Turn the heat up to medium under your frying pan, garlic still in. Pour in a dash of white wine and let reduce until the alcohol is evaporated. Add a little cream and let the sauce thicken. Fish the garlic out of the sauce and remove the skins. Put a number (or all) back in the sauce and crush them. Add a handful of fresh finely cut tarragon (best ever herb with chicken!). Cut chicken, sauce, serve and enjoy. And, much less calories than aïoli...