Aioli: Put garlic in blender with the yolks, lemon juice and basil. Slowly add olive oil. Set aside.
Preheat oven to 400 degrees.
Pound out the chicken until thin. Place the breasts one on top of the other in an overlapping fashion. Mix the olives, artichokes, tarragon, salt and pepper together.
Place 2 pieces of the ham on top of the chicken in an overlapping manner. Then add the artichoke mixture. Sprinkle the cheddar cheese on top. Layer with a slice of the Swiss cheese. Place 2 more ham slices on top. Place remaining slice of Swiss on top of ham. Roll the chicken log and tie with the string. Chill.
Toss in seasoned flour and saute on all sides until golden brown. Place in oven and bake 25 minutes. Remove and chill.
Cut off string and slice thin. Serve with Basil aioli.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.