Brewer's: Dutch Gleek
Tippling. Gleek is a game, and the phrase means the game loved by Dutchmen is drinking.
“Nor could be partaker of any of the good cheer except it were the liquid part of it, which they call `Dutch Gleek.' ” —Gayton.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894