In a large mixing bowl, combine the ground chicken, grated onion, minced garlic, breadcrumbs, Cajun seasoning, and salt (if needed). Mix gently with your hands until just combined. Avoid overmixing, as it can make the burgers dense.
Divide the mixture into 4 equal portions and shape into patties about ½-inch thick. Since ground chicken is sticky, lightly wet your hands or coat them with a little oil to make shaping easier.
Preheat a grill or cast-iron skillet over medium heat. If using a skillet, add a small drizzle of oil.
Cook the burgers for 5–6 minutes on the first side, then carefully flip and cook another 4–6 minutes, or until the internal temperature reaches 165°F.
During the last 1–2 minutes of cooking, place a slice of cheddar cheese on each burger and cover the skillet or close the grill lid until the cheese melts.
While the burgers finish cooking, place the buttered buns cut-side down on the grill or skillet and toast for about 1 minute until golden.
In a small bowl, whisk together the mayonnaise, chipotle sauce, and lime juice until smooth.
Spread the chipotle mayo on both halves of each toasted bun. Layer with lettuce, the cheesy chicken burger, sliced red onion, and the top bun.
Serve immediately with crispy French fries or a fresh side salad.