Preheat a grill or grill pan to medium-high heat. Season the chicken thigh evenly with salt. Lightly oil the grill rack or brush the grill pan with oil.

Grill the chicken, turning once, until fully cooked through, approximately 8 to 12 minutes. Remove from the grill and allow the chicken to rest for 5 minutes. Shred the chicken using two forks and transfer to a small bowl. Add the barbecue sauce and stir until evenly coated.

Arrange the bread slices on a work surface. Spread the softened butter evenly over one side of each slice. Turn two slices over and divide the barbecue chicken evenly between them. Top the chicken with the shredded pepper jack cheese. Spread the mashed avocado evenly over the remaining two slices of bread. Close the sandwiches, ensuring the buttered sides face outward.

Place the sandwiches in a large non-stick skillet set over medium heat. Cook until the bottoms are golden brown, approximately 3 to 5 minutes. Carefully turn the sandwiches over, reduce the heat to medium-low, and continue cooking for an additional 3 to 5 minutes, or until the second side is browned and the cheese is fully melted. Remove from heat and serve immediately.