1headroasted garlic (see instructions below)or substitute chopped fresh garlic
2tsplemon juice (from half of a lemon)
milk to thin, if desired
salt & pepper to taste
Trim the stems (to about 1 inch) and about 1 inch from the tops.
Bring a gallon of salted water to a boil in a large pot. Squeeze lemon halves into water and add to water, along with trimmed artichokes. Simmer artichokes and cook until tender, about 15-20 minutes (a knife should slide through fairly easily). Remove artichokes with tongs and let any excess water drip from tops. Drain on paper towels and set aside until cool enough to handle.
Cut each artichoke in half and then in half again into quarters. Remove the hairy choke using a spoon or grapefruit spoon and discard. Set aside in a large bowl or a large gallon ziptop bag.
In a medium bowl, whisk together the balsamic vinegar, 4 cloves of garlic, rosemary, olive oil, pepper, and salt to taste. Pour over the artichokes and toss together to coat. Marinate for at least 30 minutes, but up to two days.
Prepare garlic aioli. Cut off top of head of garlic and drizzle with a little olive oil. Sprinkle with salt and pepper and wrap in foil. Bake at 400° F for 30-40 minutes, or until soft. Squeeze garlic into small bowl and use a fork to mash. Add mayonnaise, parsley, lemon juice, salt & pepper. To thin out, if needed, add a few dashes of milk and stir.
Prepare gas or charcoal grill for high heat.
Using tongs, place the artichokes on the grill and cook for about 30 seconds per side until well charred.
Remove from the grill and serve warm with garlic aioli.