Make cornbread batter: Preheat oven to 425 oF with rack in center position. Generously oil 4 muffin cups or a 6-inch skillet, and put muffin tin or skillet in oven to preheat. In a medium bowl, whisk cornbread mix, 1 tablespoon sugar, and 1⁄2 teaspoon salt. Add 2 tablespoons vegetable oil, 1⁄2 cup of the milk (save rest for own use), and a few grinds pepper. Stir to combine.
Bake cornbread: Pour cornbread batter into preheated muffin cups or skillet (batter will be runny), and bake in oven until cooked through and golden brown and crisp around the edges, about 18 minutes in muffin tin, or 12 minutes in skillet. Allow to cool in pan for 5 minutes.
Prep ingredients: Trim ends from shallot, then peel, and thinly slice about 1⁄3 cup (save rest for own use). Thinly slice cucumbers on an angle. Roughly chop dill fronds and tender stems, discarding thicker stems.
Marinate cucumbers: In a medium bowl, whisk together vinegar, 2 tablespoons oil, 1⁄2 teaspoon salt, and a few grinds pepper. Add cucumbers and shallot and toss to combine. Set aside to marinate.
Grill chicken: Heat a grill or grill pan over high. Pat chicken dry and pound to even 1⁄2-inch thickness, if necessary. Season all over with salt and pepper. Lightly oil grill grates, add chicken, and cook 2 minutes per side. Brush with BBQ sauce and cook 1 minute on each side (brushing again after flipping) until sauce is nicely caramelized and chicken is cooked through.
Finish & serve: Add dill to cucumbers and season to taste with salt and pepper. Serve cucumbers alongside BBQ chicken and cornbread. Enjoy!