Income Per Share - Schedule of Computation of Basic Income Per Share (Details) - USD ($) $ / shares in Units, shares in Thousands, $ in Thousands	3 Months Ended
	Nov. 25, 2023	Nov. 26, 2022
Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]
Net income	$ 42,325	$ 33,957
Allocation of net income for Basic:
Common stock	$ 42,325	$ 33,957
Weighted average number of shares for Basic:
Common stock (in shares)	18,702	18,672
Common Stock
Allocation of net income for Basic:
Common stock	$ 35,566	$ 28,525
Weighted average number of shares for Basic:
Common stock (in shares)	15,111	15,082
Income per share for Basic:
Common stock (in dollars per share)	$ 2.35	$ 1.89
Class B Common Stock
Allocation of net income for Basic:
Common stock	$ 6,759	$ 5,432
Weighted average number of shares for Basic:
Common stock (in shares)	3,590	3,590
Income per share for Basic:
Common stock (in dollars per share)	$ 1.88	$ 1.51