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