This term appears in the description of the solving tourney for the 4th Problem Tourney of the British Chess Magazine.
Partial “cooks" will score either two or one point according to our estimate of their respective importance.
The term is not found in Encyclopedia of Chess Problems - Themes and Terms, or in any of the early books on chess problem terms and terminology.
The term does not seem to appear in any earlier B.C.M. issues, nor does it seem to have been generally well-known at the time, as some solvers asked for a definition. The problem editor, H. J. C. Andrews, answered:
To this we reply that while a “partial cook” is always a dual the converse of this proposition by no means follows. A “partial cook” in fact can only occur on the 2nd or 3rd move of the mainplay.
The last sentenece should probably be understood in the context of the tourney which was for three-move problems only. (See British Chess Magazine 7/74 (Feb., 1887), p. 76–77 for additional details.)
A later issue suggests that the definition may still have left some questions unanswered, as points for identifying partial cooks apparently was dropped from the solving tourney.
(The use of quotes for the word “cook” suggests that the term was not a fully accepted technical term at the time—at least not by the then problem editor—and that it needed some kind of additional indication (such as quotes, italic letters, the prefix ‘so-called’ etc.) to show that the intended meaning of the word was different from the apparent one.)