Preheat the oven to 420°F. Line a baking sheet with foil and lightly spray with nonstick cooking spray.
Lay the salami slices on the prepared baking sheet and top each with equal amounts (in order) of chicken, cheese, tomato, then basil.
Fold the salami around ingredients like a taco (keeping the cheese centered) and poke a skewer through the midsection to keep it secure. Thread equal amounts of folded salami onto the six skewers, with 1⁄4-inch space between each salami so they are not touching.
Arrange the skewers on the baking sheet spaced 1 inch apart and bake for 10 minutes. Serve immediately.