In a small saucepan, bring 1 cup water to a boil. Stir the salt and brown sugar into the boiling water until dissolved. Place the ice cubes in a large container. Pour the brine over the ice cubes and stir to mix and cool. Add the leg quarters to the cooled brine, adding additional water if needed to make sure they are submerged. Refrigerate overnight.
Remove the chicken from the brine, rinse and pat dry. Preheat the grill to medium-high. Turn one side of the grill off and place the chicken skin-side down over the indirect heat. Cook for 10 minutes, turn, cook another 10 minutes and then brush with a thick coating of basting sauce. Turn, cook another 10 minutes, baste and cook a final 10 minutes for a total of 40 minutes on the grill. The chicken should register at least 160 degrees F and the juices should run clear. Serve immediately.
In a small bowl, whisk together the egg yolks, 1/4 cup lemon juice, honey and salt. While constantly whisking, very slowly drizzle in the canola oil, drop by drop, until an emulsion forms, and then continue adding the oil in a slow steady stream. The mixture should thicken to the consistency of mayonnaise. Whisk in a little water to thin the final sauce to the consistency of a creamy salad dressing. Taste and add more salt or lemon juice, if necessary.