Place artichokes into a large pot and add remaining ingredients, except olive oil. Stir and bring mixture to a boil. Once boiling, reduce heat to medium-low and simmer for 30 to 40 minutes or until artichoke hearts, in each half, are fork tender.
Remove artichokes from liquid and brush with oil. Season with salt and pepper.
Place halves on grill, cut side down, and grill for 3 to 4 minutes. Flip and continue to grill for 2 minutes. Remove from grill and serve with aioli and butter.
For aioli: Place egg, lemon juice and mustard in a blender and pulse until smooth. With the motor running begin adding oil, in tiny, slow drips, waiting for the mixture to emulsify before adding more. Continue until all oil has been use and mixture is thick. Add roasted garlic and pulse until desired consistency has been achieved. Season with salt and pepper.
For clarified butter: Place butter in a pot and melt over medium-low heat Once fully melted the milk solids should sink. Skim foam off top and ladle clear butter into a serving container. You can also strain mixture through a coffee filter, but should press the mixture through, as you’ll press the solids through, leaving you with cloudy butter.