Mix cream cheese, 2 teaspoons dill, and 2 tablespoons garlic together in a medium bowl; set aside.

Season chicken with remaining garlic and dill.

Melt 1 tablespoon butter in a skillet over medium heat. Add chicken and cook until no longer pink in the center and the juices run clear, about 8 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove from the pan.

Butter one side of each slice of bread with remaining butter. Toast buttered bread in the skillet until golden, about 2 minutes.

Remove bread from the skillet. Spread 1/2 of the cream cheese mixture onto the unbuttered sides of 2 bread slices. Top each slice with lettuce, tomato slices, chicken breast, and remaining bread.