The curious etymology of "pie"
Solution 1:
I don't have an answer but, there are a few French idiomatic expressions in which "pie" is connected to food. I mention them below in case somebody knows more pieces of the jigsaw puzzle.
Pie is the French word for Magpie. The "Mag" part is an abbreviation of Margot because this is also one of the various dialectal names of the Magpie in French.
There is an old French expression for a ricotta like young cheese: Fromage à la pie. From here the interpretations vary. Some say that this cheese was black and white because it was mixed with herbs (I find this interpretation dubious) and others say that this kind of fresh cheese is the best bait to tame a young magpie.
Another French expression closer to your citation is faire une pie de mouton. This expression refer to the recipe of broiling mutton blade-bones in order to strip off the last bits of meat. The link between the Magpie and the pie is still absent but, at least we're in the domain of food.
Finally, there is considerable evidence that the pie was introduced in Europe by returning crusaders from Middle Eastern countries. The early mince pie was also known as Christmas Pye, Crib Pye (because there was a place for the Christ Child on top of it) or Manger Pye ("manger" being French for "to eat"). The original pie was filled with various meats such as chicken, partridge, pigeon, pheasant, rabbit, etc.
[All these phrases are easily found in ancient French and English dictionaries accessible on Google books.]
Solution 2:
It seems English experts everywhere are confused on this but the general consensus seems to be that the food "pie" derives from "magpie"
Word Detective is one of many links claiming the same
The second sort of “pie” is the familiar dish, a pastry shell filled with an assortment of ingredients, usually including either meat or fruit. This “pie” appeared in English several hundred years after “pie” meaning “magpie,” and opinions vary as to the origin of this “pie.” But the first edible “pies” were a jumble of meat and vegetables, reminiscent of a magpie’s trove of odd objects, making it probable, in the view of many authorities, that the two “pies” are actually the same word. The bird’s quirky housekeeping, in short, gave us our modern “pie.”
And from WorldWideWords
One historian of the language has suggested that the food was named after the bird because the varied ingredients reminded people of the birds’ habit of collecting together all sorts of bits and pieces in their nests. Nobody’s been able to prove or disprove this thesis, and after all this time it’s unlikely anybody ever will.
This is not definitive, as it can just as likely be a story made up to fill the gap.
The characteristic feature of pies in the Middle Ages was that their filling consisted of a heterogeneous mixture of ingredients (as opposed to pastries, which had just one main ingredient). This has led etymologists to suggest that pies were named after magpies (or pies, as they were originally called), from a supposed resemblance between the miscellaneous contents of pies and the assortment of objects collected by thieving magpies.
Solution 3:
I quoted the song as a far fetched joke, but then I found the following under article for pie bird:
The nursery rhyme "Sing a Song of Sixpence" refers to "Four and twenty blackbirds, baked in a pie; when the pie was opened, the birds began to sing" but it is uncertain whether pie vents were designed to look as birds because of this song. The Oxford English Dictionary comments that the word pie itself (in the culinary sense) may be connected with 'pie' as the name of a variety of birds, in particular the magpie, and also comments on a putative relationship between the similar terms haggis and haggess (another obsolete name for a magpie).
Taking into consideration that earliest recorded reference to pies goes to 9500 BC Egypt and that the word for pie crust was Gk. kofinus, L. cophinus “basket” and that this was principally a container for baking and everything baked was a pie there must have been a myriad of terms that describe a pie with very unclear and intertwined etymology.
The link to magpie could be substantiated through another lane - as etymology of pie from word-origins states,
pie has now been superseded by magpie as the bird-name, it survives in pied (14th c.) (etymologically ‘coloured black and white like a magpie’) and piebald (16th c.) (etymologically ‘streaked with black and white’).
In both cases the typical black and white pattern is found in these images - a possible reason to call a pie a pie as the crust might have been white and the insides dark.
Another fact is that the bird name moved to magpie (by adding Mag, short of Margaret, see here) - probably due to clashing of meanings. However, this does not help etymologically the meanings might have clashed regardless of the fact if the etymologies were shared or not.