Step 1

Combine 4 cups water with the sugar, ginger, and 1/4 cup salt in a large pot. Halve the lemon, squeeze the juice from half and add the juiced half. Reserve the remaining lemon half. Bring the water to a simmer over medium-high heat and stir until the salt and sugar are dissolved. Remove from the heat, add the tea bags and let steep for 5 minutes. Remove the tea bags and add 3 cups of ice to cool the mixture to room temperature, 5 to 8 minutes. Add the chicken pieces and make sure they are submerged. Cover and refrigerate for at least 8 hours and up to 16 hours.

Step 2

Remove the chicken from the fridge and remove from the brine. Pat dry.

Step 3

Prepare a grill for medium indirect heat: For gas grills (with 3 or more burners), turn all the burners to medium-high heat; after about 15 minutes turn off one of the middle burners (2 if you have a grill with 4 or more burners) and turn the remaining burners down to medium. For charcoal grills, bank one chimney starter-full of lit and ashed-over charcoal briquettes to one side of the grill. Set up a drip pan on the other side to avoid flare-ups. The grill should be around 350 degrees F so adjust the burners or air-flow vents as necessary to maintain that temperature. (Be sure to consult the grill manufacturer’s guide for the best results.)

Step 4

Lightly oil the grates of the grill and lay the chicken pieces, bone-side down, on the indirect heat side of the grill. Cover and cook until the skin is golden and the chicken is cooked about three-quarters of the way (130 to 135 degrees F on an instant-read thermometer), 20 to 25 minutes. Flip the chicken pieces so they are skin-side down and continue indirect grilling until the chicken is cooked through and reads 165 degrees F on an instant-read thermometer, about 15 minutes more.

Step 5

Meanwhile, lightly oil the flesh of the peaches and the remaining lemon half and place flesh-side down on the direct side of the grill. Grill until marked but not too soft, about 2 minutes. Turn the peaches (leave the lemon flesh-side down) and grill just until marked on the other side, about 1 minute more. Remove the peaches and lemon from the grill. Slice the peaches into wedges. Squeeze the lemon into a small bowl and whisk in the olive oil. Season with salt and pepper. Toss the arugula with the mint, peaches and the grilled-lemon vinaigrette. Serve with the chicken.