The Beer, the Walk, the Usage

Note: this ancient joke depends upon popular familiarity with Miller High-Life Beer's slogan, "The beer that made Milwaukee famous."

     The home team's pitcher, Milt Famee, was in the habit of drinking a beer between the innings he pitched.  One famous tied-score game went into six overtime innings, and by the fifteenth inning, Milt was so sozzled that he walked four successive batters, walking in the winning run.  Members of the visiting team were heard to praise "the beer that made Milt Famee walk us."

That: if it's "the beer that," it was the fifteenth beer. 

Which: if it's "the beer which," it was all that beer, cumulatively.