Make brine by combining water, sugar and salt in large stockpot. Bring to a boil over high heat. Reduce heat to medium-low and simmer 10 minutes, or until sugar and salt are dissolved. Remove from heat and cool completely. Butterfly chicken by removing backbone by cutting down either side of the backbone with a sharp knife. Remove the backbone. Turn chicken over so that rib bones are facing up and cut out breastbone. When brine is cool, add chicken to stockpot and refrigerate at least one, and up to five, hours. Do not brine longer as chicken will become too salty. Remove chicken from brine and pat with paper towels to dry.
Heat grill to high, or, if using coals, heat and arrange with more coals on one side of grill than the other. (This creates two areas of heat – one high and the other cooler.) Place chicken over high heat and grill about 4 minutes per side until nicely browned.
Turn off middle burner of gas grill (or one burner if there are only two) and move chicken over the area that is turned off. Or, if using a charcoal grill, move chicken to cooler area of grill. Grill chicken for another 18 – 20 minutes, turning often, until chicken registers 180 degrees F when a thermometer is inserted into the thickest part of the thigh.
While chicken is grilling, make vinaigrette by combining all ingredients except the oil in a food processor. Pulse to combine. With motor running, slowly pour oil into mixture; continue pulsing until well-incorporated. Remove chicken to a platter and pour vinaigrette over hot chicken. Turn chicken to coat in sauce.