Purée cilantro, garlic, jalapeño, lime juice, and salt in food processor until minced. Slowly drizzle in olive oil while the machine is running, forming a paste.
Mix mayonnaise, chipotle purée, and sugar in a small bowl.
Spread each slice of bread with butter or plain mayonnaise on one side. Spread the cilantro pesto on the other side of 2 slices and top with cheese and chicken.
Spread the other 2 slices with the chipotle mayo and place on top of the chicken, buttered side up.
Toast on a panini press or in a skillet, pressing firmly or placing a weighted item on top. Cook over medium heat. When the first side is browned, flip over and cook the other side
Notes
If canned chipotle chili purée is unavailable, substitute for 1/2 tablespoon of chipotle adobo paste.