Mix all ingredients together in a saucepan. Bring to a boil, stirring until sugar and salt are dissolved; cool.
For the Chicken
French the wing joint of each chicken breast and place in large bowl. Pour brine over chicken; cover and refrigerate at least 2 hours or up to 12 hours. Remove chicken from brine and pat dry; discard brine.
Heat skillet over medium-high heat until hot; add 2 tablespoons oil and chicken. Sear chicken 2 minutes per side or until browned; rub chicken with garlic, thyme and rosemary. Transfer skillet to oven and roast at 375°F for 15 minutes (350°F for convection) or until internal temperature reaches 165°F. Remove chicken from skillet; reserve.
For the jus, add the chicken stock and remaining 2 tablespoons oil to the skillet; heat to simmering over medium-high heat. Simmer 5 minutes, scraping bottom of pan to release flavors into jus.
For the Fingerling Vegetables
Toss potatoes with 3 tablespoons oil and 2½ teaspoons salt and pepper. Roast potatoes at 375°F for 12 minutes (350°F for convection) or until blistered and browned.
Into 4 small vacuum bags, evenly distribute carrots, asparagus, green beans, butter, thyme and ½ teaspoon salt; seal bags. Cook vegetables in immersion circulator at 190°F for 10 minutes. Hold at 140°F up to 20 minutes.
Place vegetables in bowl. Arrange vegetable bowl, chicken and potatoes on plate; drizzle chicken and potatoes with jus and garnish with fresh thyme.