Income Per Share - Schedule of Computation of Diluted Income Per Share (Details) - USD ($) $ / shares in Units, shares in Thousands, $ in Thousands	3 Months Ended		6 Months Ended
	Feb. 24, 2024	Feb. 25, 2023	Feb. 24, 2024	Feb. 25, 2023
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 20,457	$ 17,810	$ 62,782	$ 51,767
As reported - Basic (in shares)	18,696	18,677	18,700	18,674
As reported - Diluted	$ 20,457	$ 17,810	$ 62,782	$ 51,767
As reported - Diluted (in shares)	18,754	18,767	18,758	18,757
As reported - Diluted (in dollars per share)	$ 1.09	$ 0.95	$ 3.35	$ 2.76
Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 17,188	$ 14,962	$ 52,754	$ 43,488
As reported - Basic (in shares)	15,106	15,087	15,110	15,084
As reported - Basic (in dollars per share)	$ 1.14	$ 0.99	$ 3.49	$ 2.88
Add: effect of dilutive potential common shares	$ 0	$ 0	$ 0	$ 0
Add: effect of dilutive potential common shares (in shares)	58	90	58	83
As reported - Diluted (in dollars per share)	$ 1.09	$ 0.95	$ 3.35	$ 2.76
Class B Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 3,269	$ 2,848	$ 10,028	$ 8,279
As reported - Basic (in shares)	3,590	3,590	3,590	3,590
As reported - Basic (in dollars per share)	$ 0.91	$ 0.79	$ 2.79	$ 2.31