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Transactions with Related Parties (Details) $ in Thousands	3 Months Ended
Transactions with Related Parties (Details) $ in Thousands	Apr. 02, 2023 USD ($) number_of_restaurants	Apr. 03, 2022 USD ($)	Jan. 01, 2023 USD ($)
Related Party Transaction
Number of Restaurants \| number_of_restaurants	7,095
TimWen \| Franchise Rental Expense
Related Party Transaction
Related Party Transaction, Expenses from Transactions with Related Party	$ 4,673	$ 4,350
TimWen \| General and administrative \| Management Fee Income
Related Party Transaction
Related Party Transaction, Other Income from Transactions with Related Party	60	54
Yellow Cab \| Royalty, Advertising Fund, Lease and Other Income
Related Party Transaction
Related Party Transaction, Other Income from Transactions with Related Party	3,610	$ 3,067
Yellow Cab \| Accounts and notes receivable, net and Advertising funds restricted assets \| Royalty, Advertising Fund, Lease and Other Income
Related Party Transaction
Accounts Receivable, Related Parties, Current	$ 1,271		$ 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, Expenses from Transactions with Related Party	$ 419
AMC \| Advertising funds restricted liabilities \| Advertising Funds Expense
Related Party Transaction
Accounts Payable, Related Parties, Current	$ 450