Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet.
Place chicken breasts into a large pot and cover with salted water by several inches; bring to a boil. Reduce heat to medium and simmer until no longer pink in the center and the juices run clear, 15 to 20 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Remove chicken from water and cool. Shred chicken using two forks.
Thoroughly mix Monterey Jack cheese, Cheddar cheese, cream cheese, salt, and pepper together in a bowl. Stir chicken into cheese mixture.
Separate crescent roll dough and lay each piece flat on a work surface.
Spread chicken-cheese mixture onto each piece of dough and roll dough into crescent roll shapes.
Place rolls on prepared baking sheet and brush with melted butter.
Bake in the preheated oven until cheese is melted and rolls are golden, 20 to 25 minutes.
