How to Make It
Prepare a grill for indirect medium- high heat. Combine brown sugar, paprika, 1 teaspoon salt, and ½ teaspoon pepper. Brush chicken with 1 tablespoon oil; season with brown sugar mixture.
Place chicken, skin side down, over direct heat. Grill until lightly charred, 5 to 10 minutes. Flip and transfer to indirect heat. Cover and cook until a thermometer inserted in thickest part registers 165°F, 35 to 40 minutes.
Meanwhile, toss tomatoes with cucumber, lemon juice, and remaining 1 tablespoon oil, 1 teaspoon salt, and ½ teaspoon pepper. Top with basil and serve with chicken.