Related Parties (Details) $ in Thousands		3 Months Ended		12 Months Ended
	Apr. 30, 2019	Jan. 01, 2023 USD ($) Rate shares	Jan. 02, 2022 USD ($) shares	Jan. 01, 2023 USD ($) yr Rate shares	Jan. 02, 2022 USD ($) shares	Jan. 03, 2021 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. 30, 2018
Related Party Transaction [Line Items]
Preferred Stock, Shares Issued | shares		100	100	100	100
Convertible Preferred Stock, Common Shares Issuable upon Conversion | shares		9,414,580		9,414,580
Franchise Term | yr				20
Franchise agreement, term				20 years
Related Party Transaction, Accounts Payable		$ 30,491	$ 31,164	$ 30,491	$ 31,164
Monthly Digital Platform Fee		$ 2,100	$ 1,300
Area Development Agreement, Restaurants To Be Opened, Built, And Operated | restaurant										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				$ 125,481	122,662	118,444
Scenario, Forecast | Subsequent Event
Related Party Transaction [Line Items]
Area Development Agreement, Restaurants To Be Opened, Built, And Operated | restaurant							12	12	12
Related Party, Burger King Corporate [Member]
Related Party Transaction [Line Items]
Related Party Transaction, Royalty Fee Rate | Rate		4.50%		4.50%
Royalty Expense				$ 76,800	72,800	67,200
Related Party Transaction, Advertising Fee Rate | Rate		4.00%		4.00%
Advertising Expense				$ 67,700	$ 64,000	59,300
Leases, Number of Leased Restaurants		217	225	217	225								232
Related Party Transaction, Accounts Payable		$ 16,000	$ 16,300	$ 16,000	$ 16,300
Related Party Transaction, Percentage of Voting Interests Acquired | Rate		15.10%		15.10%
Lessee, Operating Leases, Rent Expense, Net				$ 27,700	$ 26,900	$ 25,900
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%		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		94		94
Series B Preferred Stock [Member]
Related Party Transaction [Line Items]
Preferred Stock, Shares Issued | shares		100		100
Popeyes Franchises [Member]
Related Party Transaction [Line Items]
Right of First Refusal, Number of States | shares						2