"steal game," 1520s, "to push, poke," from French pocher "to thrust, poke," from Old French pochier "poke out, gouge, prod, jab," from a Germanic source (compare Middle High German puchen "to pound, beat, knock," German pochen, Middle Dutch boken "to beat") related to poke (v.). Sense of "trespass upon another's preserves for the sake of stealing game; kill and carry off game in violation of the law" is attested from 1610s, perhaps via the notion of "thrusting" oneself onto another's property, or perhaps from French pocher "to pocket" (the property of another); see poach (v.2). Related: Poached; poaching.
"cook in boiling liquid," mid-15c. (implied in pocched egges), from Old French poché, past participle of pochier (12c.), apparently literally "put into a pocket" (perhaps as the white of an egg forms a pocket for the yolk), from poche "bag, pocket," from Frankish *pokka "bag," from Proto-Germanic *puk- (see poke (n.1)). But connection to poke (v.) via the notion of a "broken" (shell-less) egg also has been proposed.
Pochee as a noun in reference to a way of preparing eggs is attested in a late 14c. cookery book, and eyron en poche for "poached eggs" is attested from early 15c. Related: Poaching.