Preheat the oven to 400°F. Pierce the sweet potato a few times with a fork and bake for 45–50 minutes until completely tender. (Alternatively, microwave on high for 8–10 minutes.) Once cool enough to handle, peel and measure out 1 cup of flesh.
Reduce the oven temperature to 350°F and line a baking sheet with parchment paper.
Add the sweet potato flesh, chickpeas, barbecue sauce, salt, chili seasoning, and breadcrumbs to a food processor. Pulse until the mixture is uniform and holds together — stop before it becomes completely smooth. Scrape down the sides as needed.
With lightly wet hands, divide the mixture into 5 equal portions and shape into round patties about ½ inch thick. Place on the prepared baking sheet and freeze for 15 minutes to firm up.
Bake for 30 minutes, flipping carefully at the 15-minute mark, until the patties are firm and lightly golden on both sides. Let cool for 5 minutes before serving.
Toast the hamburger buns. Spread vegan mayo on the inside of each bun, then stack with a patty, lettuce, tomato, and pickles. Serve immediately.