Heat olive oil in a skillet over medium heat and sauté sliced onions until caramelized, approximately 12 minutes.
In a small bowl, whisk together mayonnaise, adobo sauce, and minced chipotle pepper to create the chipotle aioli.
In a separate bowl, toss the shredded chicken with half of the chipotle aioli, salt, and pepper.
Spread softened butter on one side of each sourdough bread slice.
Place two slices of bread, butter-side down, on a room-temperature griddle or skillet.
Spread the remaining chipotle aioli on the upward-facing side of the bread slices.
Layer sharp cheddar cheese, the chipotle chicken mixture, caramelized onions, Monterey Jack cheese, and the remaining bread slices with the butter-side facing up.
Turn heat to medium-low and grill until the bottom bread is golden brown and crispy, approximately 4 minutes.
Carefully flip the sandwiches and grill the second side until the cheese is fully melted and the bread is toasted, approximately 3-4 minutes.
Remove from heat, allow to rest for 60 seconds to set the cheese, and serve.