First, make the chipotle chicken. Add sliced chicken to a small frying pan and season with chipotle powder, paprika, garlic powder, onion powder, and salt. Cook over medium-high heat until cooked through, about 6 to 8 minutes, stirring occasionally.
Transfer cooked chicken to a small bowl and set aside. Add vegetable oil to the frying pan, lower heat to medium, and add sliced bell pepper and onion. Sauté until vegetables are slightly softened, about 2 to 3 minutes, then remove from heat.
Preheat a cast iron griddle over medium-low heat. Spread 1 tsp butter on the outside of each slice of sourdough bread, then spread 1 tsp chipotle mayo on the inside of each slice of bread.
Assemble the chipotle chicken melt. Place one slice of bread butter side down (chipotle mayo side up) on the griddle. Top with one slice of pepper jack cheese, then add the cooked chipotle chicken, then the peppers and onions. Top with remaining slice of cheese and close the sandwich with the other slice of bread, chipotle mayo side down.
Cook until the bottom turns golden brown and the cheese is melted, about 3 to 4 minutes, then carefully flip the sandwich and cook until the bottom is golden brown, another 3 to 4 minutes. Optionally, you can use a preheated cast iron grill press on top of the sandwich to cook from both sides at the same time and avoid flipping the sandwich.
Transfer chipotle chicken melt to a cutting board, slice in half, and serve immediately.