Remove the butter from the refrigerator to soften. Preheat the oven to 450°F. Wash and dry the fresh produce. Halve the buns. Quarter the cucumbers lengthwise, then halve crosswise. In a bowl, combine the cucumber pieces, white balsamic vinegar, peppers, and a drizzle of olive oil; season with salt and pepper. Set aside to marinate, stirring occasionally. In a separate bowl, combine the carrots and apple cider vinegar; season with salt and pepper. Stir to combine. In a separate bowl, combine the barbecue sauce, worcestershire sauce, 1/4 cup of water (or 1/2 cup for 4 servings), and as much of the chile paste as you'd like, depending on how spicy you'd like the dish to be. In a separate bowl, combine the honey and half the softened butter; season with salt and pepper. Using a fork, mash to thoroughly combine.

In a bowl, whisk together the cornbread mix, baking powder, and a pinch of salt. Melt the remaining softened butter in a large bowl in the microwave (or melt in a small pot on the stove, then transfer to a large bowl). Add the sour cream, buttermilk, and egg(s); whisk to thoroughly combine. Add the dry ingredients to the bowl of wet ingredients. Stir until just combined (be careful not to overmix).

Lightly grease a loaf pan (or an 8-inch square baking dish for 4 servings). Pour the batter into the pan and spread into an even layer. Bake 11 to 13 minutes, or until set and cooked through (a toothpick inserted should come out with a few moist crumbs). Remove from the oven.

Meanwhile, in a medium nonstick pan (or large for 4 servings), heat a drizzle of olive oil on medium-high until hot. Working in batches if necessary, add the halved buns, cut side down. Toast 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface. Wipe out the pan.

In the same pan, heat a drizzle of olive oil on medium-high until hot. Add the pulled chicken (tearing the pieces apart before adding). Cook, stirring occasionally, 1 to 2 minutes, or until lightly browned. Add the spicy BBQ sauce; season with salt and pepper. Cook, stirring frequently, 1 to 2 minutes, or until combined and the chicken is coated and heated through. Turn off the heat. Assemble the sandwiches using the toasted buns, glazed chicken, and carrot slaw. Serve the sandwiches with the baked cornbread and marinated cucumbers on the side. Top the cornbread with the honey butter. Enjoy!