Income Per Share - Schedule of Computation of Diluted 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 Diluted [Line Items]
As reported - Basic	$ 42,325	$ 33,957
As reported - Basic (in shares)	18,702	18,672
As reported - Diluted	$ 42,325	$ 33,957
As reported - Diluted (in shares)	18,769	18,754
As reported - Diluted (in dollars per share)	$ 2.26	$ 1.81
Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 35,566	$ 28,525
As reported - Basic (in shares)	15,111	15,082
As reported - Basic (in dollars per share)	$ 2.35	$ 1.89
Add: effect of dilutive potential common shares	$ 0	$ 0
Add: effect of dilutive potential common shares (in shares)	67	82
As reported - Diluted (in dollars per share)	$ 2.26	$ 1.81
Class B Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 6,759	$ 5,432
As reported - Basic (in shares)	3,590	3,590
As reported - Basic (in dollars per share)	$ 1.88	$ 1.51