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Transactions with Related Parties (Details) $ in Thousands	6 Months Ended
Transactions with Related Parties (Details) $ in Thousands	Jul. 02, 2023 USD ($) number_of_restaurants	Jul. 03, 2022 USD ($)	Jan. 01, 2023 USD ($)
Related Party Transaction
Number of Restaurants \| number_of_restaurants	7,115
Accounts payable	$ 38,640		$ 43,996
TimWen \| Franchise Rental Expense
Related Party Transaction
Operating Costs and Expenses	10,140	$ 9,601
TimWen \| Management Fee Income \| General and administrative
Related Party Transaction
Other Operating Income	121	111
Yellow Cab \| Royalty, Advertising Fund, Lease and Other Income
Related Party Transaction
Other Operating Income	7,411	$ 6,476
Yellow Cab \| Accounts and notes receivable, net and Advertising funds restricted assets \| Royalty, Advertising Fund, Lease and Other Income
Related Party Transaction
Accounts receivable, net, current	$ 1,130		$ 1,125
Yellow Cab \| Franchised Units
Related Party Transaction
Number of Restaurants \| number_of_restaurants	83
AMC \| Advertising Funds Expense
Related Party Transaction
Related Party Transaction, Purchases from Related Party	$ 1,200
AMC \| Advertising funds restricted liabilities \| Advertising Funds Expense
Related Party Transaction
Accounts payable	$ 850