Stir mayonnaise, chipotle, lime juice, garlic powder, cumin, salt, and pepper in a bowl until smooth.
Fold in the shredded chicken until evenly coated with the chipotle-lime mayo.
Spread softened butter on one side of each sourdough slice.
Warm a large skillet or griddle over medium heat.
Assemble sandwiches on the unbuttered sides: smear a little extra chipotle mayo, add cheese, chicken, avocado, onion, and cilantro, then top with more cheese and a bread slice, buttered side out.
Cook 3–4 minutes per side until bread is deep golden and cheese melts, pressing gently with a spatula.
Rest 1 minute, then slice and serve hot.