"a German" (especially a German soldier), 1841, but popularized during World War I, from German kraut "cabbage," considered a characteristic national dish. The "cabbage" sense is attested in English from 1855.
Entries linking to kraut
sauerkraut n.
"a favorite German dish consisting of cabbage cut fine, pressed, salted, and fermented until sour," 1630s, from German Sauerkraut, literally "sour cabbage," from sauer "sour" (from Proto-Germanic *sura-; see sour (adj.)) + Kraut "vegetable, cabbage," from Old High German krut, from Proto-Germanic *kruthan.
They pickle it [cabbage] up in all high Germany, with salt and barberies, and so keepe it all the yeere, being commonly the first dish you have served in at table, which they call their sawerkrant. [James Hart, "Klinike, or the diet of the diseased," 1633]
In U.S. slang, figurative use for "a German" dates from 1858 (compare kraut). "The effort to substitute liberty-cabbage for sauerkraut, made by professional patriots in 1918, was a complete failure." [Mencken]. French choucroute (19c.) is the German word, but via Alsatian German surkrut but with folk etymology alteration in French based on chou "cabbage" + croûte "crust" (n.).