Gently simmer broth in saucepan; add chicken and poach 10 to 15 minutes to at least 165°F. Remove; shred and reserve 1 1/2 cups hot broth.
Meanwhile, cook cavatappi with tablespoon salt to 1 minute short of package time. Reserve 1 cup pasta water; drain.
Heat oil in large skillet over medium-low; stir flour 1 minute. Add chili powder, paprika, cumin, garlic powder, oregano, and paste; stir 30 seconds.
Gradually whisk in reserved broth; simmer 4 to 5 minutes until smooth. Add beans, chicken, pasta, sauce salt, and 1/4 cup pasta water; toss over low 2 minutes.
Remove from heat; melt Monterey Jack into sauce, loosening with pasta water if needed. Serve while sauce flows around noodles.