Cook rotini in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season chicken with its salt and pepper. Heat oil in a large skillet and cook chicken and celery for 6 to 8 minutes, stirring, until the meat reaches at least 165°F.
Stir in garlic for 30 seconds, then add broth, hot sauce and cream cheese. Reduce heat to low and stir for 2 to 3 minutes until the cream cheese melts smoothly.
Add rotini and 1/4 cup reserved pasta water and toss over low heat for 1 to 2 minutes until the pasta is tender.
Remove from heat, stir in Monterey Jack and add more reserved water if needed for a light coating. Serve while the sauce is smooth.