Heat a grill to high. Sear chicken breasts on each side for 3 minutes before moving to a lower temperature area of the grill. Brush with barbecue sauce and cook for 3 minutes, flip and brush other side with barbecue sauce and cook another 2-3 minutes until cooked through but still moist.

While the chicken is cooking, toss the vegetables and grapes with olive oil. Grill asparagus and radishes for about 3 minutes in a grill basket, on a grill tray, or on a piece of aluminum foil set on the grill grates (this will prevent vegetables from falling through the grates). Add scallions, tomatoes, and grapes and grill until all the vegetables and grapes are cooked but still tender. The grapes and scallions will be more wilted, which is ok.

Remove chicken and vegetables from grill and set aside.

Divide the arugula and red lettuce among four plates. Top each plate of greens with an even mixture of grilled vegetables and grapes. Slice grilled chicken breasts and divide among plates.

Whisk together lemon juice, olive oil, mint, garlic, salt, and pepper. Drizzle on salads and serve.