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Transactions with Related Parties Related Party Transaction Summary (Details) $ in Thousands		12 Months Ended
	Jan. 07, 2021 number_of_restaurants	Dec. 31, 2023 USD ($) number_of_restaurants	Jan. 01, 2023 USD ($)	Jan. 02, 2022 USD ($)	Jun. 30, 2021 USD ($) ft²	Nov. 30, 2018 USD ($) ft²
Related Party Transaction
Number of Restaurants \| number_of_restaurants		7,240
Accounts receivable, net		$ 104,797	$ 98,563
Advertising funds restricted liabilities		$ 120,558	132,307
Franchised Units
Related Party Transaction
Number of Restaurants \| number_of_restaurants		6,825
QSCC
Related Party Transaction
Proceeds from Rents Received		$ 231	198	$ 217
Area of Real Estate Property \| ft²					18,774	14,493
Annual Base Rent					$ 250	$ 217
QSCC \| Cost of sales \| Patronage Dividends
Related Party Transaction
Related Party Transaction, Purchases from Related Party		363	427	279
TimWen
Related Party Transaction
Operating Costs and Expenses		20,653	19,694	18,687
TimWen \| Franchise Rental Expense
Related Party Transaction
Operating Costs and Expenses		20,894	19,927	18,906
TimWen \| General and administrative \| Management Fee Income
Related Party Transaction
Other Operating Income		$ 241	233	219
Yellow Cab \| Franchised Units
Related Party Transaction
Number of Restaurants \| number_of_restaurants		83
Significant Changes, Franchises Purchased During Period \| number_of_restaurants	54
Yellow Cab \| Royalty, Advertising Fund, Lease, and Other Income
Related Party Transaction
Other Operating Income		$ 14,757	13,404	9,869
Yellow Cab \| Accounts Receivable and Advertising Funds Restricted Assets \| Royalty, Advertising Fund, Lease, and Other Income
Related Party Transaction
Accounts receivable, net		1,153	1,125
AMC \| Advertising Funds Expense
Related Party Transaction
Related Party Transaction, Purchases from Related Party		2,366	$ 0	$ 0
AMC \| Advertising funds restricted liabilities \| Advertising Funds Expense
Related Party Transaction
Advertising funds restricted liabilities		$ 584