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Stock-based Compensation - Narrative (Details)
12 Months Ended
Feb. 09, 2022
member
shares
Sep. 30, 2022
USD ($)
$ / shares
Sep. 30, 2021
USD ($)
Sep. 30, 2020
USD ($)
Feb. 07, 2022
shares
Share-Based Payment Arrangement [Abstract]          
Number of shares authorized for issuance (in shares) | shares         300,000
Additional shares authorized for issuance (in shares) | shares 50,000        
Number of members of management receiving approved grant of stock | member 6        
Stock-based compensation   $ 2,353,000 $ 0 $ 0  
Stock-based compensation tax benefit   569,000      
Unrecognized stock based compensation expense, stock options   $ 7,100,000      
Compensation cost not yet recognized, period for recognition   4 years 4 months 24 days      
Vesting period 4 years        
Vesting percentage 20.00%        
Weighted average grant date fair value (in usd per share) | $ / shares   $ 31.37      
Expiration period 5 years