Whisk together mustard, 1 tbsp lemon juice, 3 tablespoons olive oil, parsley, 1/4 teaspoons sugar, 1/4 teaspoons salt, and 1/4 teaspoons pepper in small bowl; set aside.
Whisk together remaining tablespoon lemon juice, remainin 3 tablespoons olive oil, remaining teaspoons of sugar, 1 1/2 teaspoons salt, 1/2 tsp pepper, garlic, and water in medium bowl. Place marinade and chicken in gallon-size zipper-lock bag and toss to coat; press out as much air as possible and seal bag. Refrigerate 30 minutes, flipping bag after 15 minutes.
Light large chimney starter filled with charcoal (6 qts, about 100 briquets) and allow to burn until coal are fully ignited andpartially covered with thin layer of ash, about 20 minutes. Build modified 2 leverl fire by arranging all coals over half of grill, leaving the other half empty. Position cooking grate over coals, cover grill, and heat grate until hot, about 5 minutes; scrape grate clean. Dip wad of paper towels in oil, holding wad with tongs, wipe cooking grate. Grill is ready when you can hold your hand 5" above grate for 5 minutes.
Remove chicken from bag, allowing excess marinade to drip off. Place chicken on cooler side of grill, smooth-side down, with thicker side facin coals. Cover with disposable 9x13" aluminum pan and cook until bottom of chicken just begins to develop light rill marks and is no longer translucent, 6-9 minutes. Using tongs, flip chicken and rotate so that thinner side faces coals. Cover with disposable pan and continue to cook until chicken is opaque and firm to touch and instant read thermometer inserted into thickest part of chicken registers 140 degrees, 6-9 minutes longer.
Move chicken to hotter side of grill and cook,uncovered, until dark grill marks appear, 1-2 minutes. Using tongs, flip chicken and cook until dark grill marks appear and instant read thermometer inserted into thickest part of chicken registers 160 degrees, 1-2 minutes linger. Transfer chicken to cutting board, let rest, tented with foil, 5 minutes. Slice each breast on bias into 1/4" thick slices and transfer to individual plates.
Drizzle with reserved sauce to serve.
On gas grill: Turn all burners to high and heat with lid down until very hot, about 15 minutes. Scrape grill grate with grill brush to clean. Dip wad of paper towels in oil; holding wad with tongs, wipe cooking grate. Leave primary burner turned to high and turn off other burners. Proceed with recipe, grilling with lid down and omitting disposable pan. Increase browning time by 1-2 minutes.