Preheat a frying pan over medium heat. Add the avocado oil and heat until shimmering. Add the onion and jalapeño, season with a pinch of salt and cook, stirring to prevent burning, until the onion is soft and translucent, 6 to 8 minutes. Transfer the sautéed onion and jalapeño to a bowl and let it cool.
In a large mixing bowl, combine the ground chicken, sautéed onion and jalapeño, garlic, cilantro, paprika, ½ teaspoon kosher salt, ¼ teaspoon black pepper. Use your hands to mix everything together until combine and the meat is slightly tacky. Equally divide the mixture into 4 portions, wet your hands to prevent the meat from sticking, and form into burger patties. Place the patties on a sheet pan lined with parchment paper, loosely cover with plastic wrap and refrigerate for at least 15 minutes, or up to overnight, to let them firm slightly.
Preheat your Traeger grill to 500F (475F if using a ModiFire Sear Grate). If using a regular grill, preheat to medium-high heat. Transfer the patties to the grill grate (direct heat for regular grills) and cook, undisturbed, until grill marks form and the burgers are no longer sticking to the grates, 6 to 8 minutes. Flip the patties, brush the tops with BBQ sauce and cook until grill marks from on the bottom sides and the thickest part of each patty registers 160F, around 6 minutes. Flip the burgers once more, brush with more BBQ sauce, top each patty with cheese, close the grill door and cook until the cheese has melted and the thickest part of each patty registers 165F.
Serve the chicken burgers on a lettuce bun with sliced tomato, pickles, onion and extra BBQ sauce.