Bring broth to a gentle simmer in a saucepan, add chicken and poach for 10 to 15 minutes until at least 165°F. Remove and shred, reserving 1 1/2 cups hot poaching broth.
Meanwhile, cook cavatappi in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Heat oil in a large skillet over medium-low heat and stir in flour for 1 minute. Add chili powder, paprika, cumin, garlic powder, oregano and tomato paste and stir for 30 seconds.
Gradually whisk in reserved broth and simmer for 4 to 5 minutes until smooth. Add beans, shredded chicken, pasta, sauce salt and 1/4 cup reserved pasta water and toss over low heat for 2 minutes.
Remove from heat, melt Monterey Jack into the sauce and loosen with more pasta water if needed. Serve while the red sauce still moves easily around the noodles.