Put garlic, olive oil, salt and pepper and thyme in a large gallon-sized zip top bag. Squeeze the half lemon into the bag and drop the rind into the bag. Close the bag and squish it all around to mix.2.
Add chicken breasts to the marinade bag and squish it around some more. Let them sit for at least 30 minutes or up to two days. The longer you leave it, the more intense the flavor from the garlic and lemon will be.
Put garlic cloves in a small saucepan with the olive oil. Cook on low heat for 20-30 minutes, or until the garlic cloves are soft and light brown. Strain and save the garlic oil for pastas and salad dressings!2.
Put the garlic cloves, salt, pepper and mayo in a food processor. Pulse till smooth.
Put all ingredients into a gallon-sized zip top bag and set aside to marinate while the chicken marinates.2.
After you pull your chicken off the grill, turn the grill up to high and place the onions (minus the liquid) on the grates and cook for 3-4 minutes on each side. You want the onions cooked but not super limp.
Preheat the grill to 375 to 400 degrees, if using a gas grill. If not, preheat a grill pan or cast-iron skillet to high heat.2.
Remove chicken breasts from the marinade and place on oiled grill or skillet. Cook for 5 minutes, or until a nice caramelized crust has formed. Flip the breasts and reduce the heat to medium.3.
Continue to cook for an additional 5-7 minutes until done. For the last 2 minutes, put ¼ cup of blue cheese on each breast and put a lid on the pan (or close the lid of the grill) to melt the cheese.4.
When you take your chicken off the grill, turn the heat up to high to cook the red onion.
Assemble the burger:
Slather 2-3 tablespoons of the aioli on the bottom and top buns. Lay your cooked, cheesed-up chicken breast on top. Place one or two slices of tomato on top, a couple of red wine onion rings, then a couple leaves of the Bibb leaf lettuce. Add the top bun and serve!