In a medium skillet, heat oil over medium heat. Add onion and jalapeño; cook until softened and fragrant, 2 to 3 minutes. Transfer to a small bowl and let cool completely.
In a large bowl, combine turkey, onion mixture, salt, pepper, and 2 tablespoons Chipotle Mayonnaise. Divide into 4 portions; shape each portion into a 1-inch-thick patty.
Heat a 12-inch cast-iron grill pan over medium-high heat. Brush cut sides of buns evenly with ¼ cup Chipotle Mayonnaise. Working in batches, place buns cut side down on grill pan and toast until golden brown, 1 to 2 minutes.
Add turkey patties to grill pan. Cook for 3 minutes; turn and top each patty with 1 slice cheese. Cook until a meat thermometer inserted into center registers 160°, 2 to 3 minutes more.
Transfer burgers to bottom halves of toasted buns; top each with sliced tomato and avocado. Combine shredded cabbage and remaining ¼ cup Chipotle Mayonnaise, tossing to coat. Divide cabbage mixture evenly on top of avocado. Top with remaining bun halves. Serve immediately.
Chipotle Mayonnaise
In a small bowl, stir together mayonnaise and chipotles in adobo. Use immediately.
Recipe by Louisiana Cookin' at https://louisianacookin.com/southwest-turkey-burger/