Related Parties (Details) $ in Thousands		1 Months Ended	12 Months Ended
	Apr. 30, 2019	May 31, 2023 USD ($)	Dec. 31, 2023 USD ($) yr Rate shares	Jan. 01, 2023 USD ($) restaurant shares	Jan. 02, 2022 USD ($) shares restaurant	Sep. 30, 2025 restaurant	Sep. 30, 2024 restaurant	Sep. 30, 2023 restaurant	Sep. 30, 2022 restaurant	Sep. 30, 2021 restaurant	Jan. 04, 2021 state restaurant	Dec. 29, 2019	Dec. 31, 2012 shares
Related Party Transaction [Line Items]
Preferred Stock, Shares Issued | shares			100	100									100
Convertible Preferred Stock, Common Shares Issuable upon Conversion | shares			9,414,580										9,414,580
Franchise Term | yr			20
Board of directors, number of members			2									2
Advertising fund increase			0.0050
Franchise agreement, term			20 years
Related Party Transaction, Accounts Payable			$ 37,797	$ 30,491
Monthly Digital Platform Fee			3,000	2,100	$ 1,300
Area Development Agreement, Restaurants To Be Opened, Built, And Operated | restaurant								12	10	4	50
Area Development Agreement, Percentage Of New Restaurants To Be In Kentucky, Tennessee And Indiana											80.00%
Area Development Agreement, Number Of Restaurants To Be Acquired | restaurant											500
Area Development Agreement, Number Of States | state											16
Area Development Agreement, Prepaid Franchise Fees					600
Lessee, Operating Leases, Rent Expense, Net			128,804	$ 125,481	122,662
Area Development Agreement, Restaurants Completed | restaurant				6
Burger King Corporate [Member]
Related Party Transaction [Line Items]
Technology Equipment, Right Of Use		$ 12,200	8,400
Repairs, Replacements, and Improvements Provided			$ 13,100
Scenario, Forecast | Subsequent Event
Related Party Transaction [Line Items]
Area Development Agreement, Restaurants To Be Opened, Built, And Operated | restaurant						12	12
Related Party, Burger King Corporate [Member]
Related Party Transaction [Line Items]
Related Party Transaction, Royalty Fee Rate | Rate			4.50%
Royalty Expense			$ 83,900	$ 76,800	72,800
Related Party Transaction, Advertising Fee Rate | Rate			4.00%
Advertising Expense			$ 73,800	$ 67,700	$ 64,000
Leases, Number of Leased Restaurants			217	217	225
Related Party Transaction, Accounts Payable			$ 16,900	$ 16,000
Related Party Transaction, Percentage of Voting Interests Acquired | Rate			14.70%
Lessee, Operating Leases, Rent Expense, Net			$ 29,400	27,700	$ 26,900
Affiliated Entity [Member]
Related Party Transaction [Line Items]
Franchise Fees, Initial, Net			$ 50
Affiliated Entity [Member] | Burger King Corporate [Member]
Related Party Transaction [Line Items]
Notes Payable, Related Parties, Maximum Borrowing Capacity				$ 120,000
Popeyes Franchises [Member]
Related Party Transaction [Line Items]
Right Of First Refusal, Number Of Franchises | restaurant					80
Right of First Refusal, Term	6 years
Related Party, PLK
Related Party Transaction [Line Items]
Related Party Transaction, Royalty Fee Rate | Rate			5.00%
Property Leases Identical to BKC's Lease with Third Party [Member] | Affiliated Entity [Member]
Related Party Transaction [Line Items]
Leases, Number of Leased Restaurants			95
Series B Preferred Stock [Member]
Related Party Transaction [Line Items]
Preferred Stock, Shares Issued | shares			100
Popeyes Franchises [Member]
Related Party Transaction [Line Items]
Right of First Refusal, Number of States | shares					2