Normans vs. Saxons: cow = beef, sheep = mutton, chicken =?
Solution 1:
According to this paper (PDF) from Phillip Slavin, chicken was one meat that even the peasantry could afford to eat.
Although poultry occupied the smallest part of demesne livestock, constituting about two per cent of it (Table 1), its social importance and omnipresence cannot be understated, despite scholarly marginalization. Chicken meat constituted an important part of everyday diet, and it was afforded by virtually every social stratum both in England and the Continent
Since eating it wasn't reserved to rich French-speaking folk, it makes sense that the word for it as food wasn't either.
As another answer also pointed out, the word poultry came into the language from Old Norman (presumably with the Conquest).
So, given a social situation where everyone was eating chicken but only Old English speakers were raising them, it makes sense that we ended up with two words for it as food, and one (Old English-derived) word for it as an animal.
Solution 2:
Fowl and poultry are both used for birds-as-food, although not exclusively. It sounds a bit fancy. So I would say that your assumption is mistaken, because fowl is the normal word for the flesh of birds used for food.
The OED’s sense 4a of fowl is indeed “The flesh of birds used for food,” as in the phrase fish, flesh, and fowl. You would probably find it on menus more often if it weren’t so easily confused for foul, which is why poultry usually gets used instead.
However, fowl is Germanic and is related to Dutch vogel, while poultry is the word that comes to us from the Norman invaders er Conquest. A related English word is pullet, which is pronounced /ˈpʊlɪt/. It comes to us straight from French poulet, and per the OED means:
A young (domestic) fowl, between the ages of chicken and mature fowl; but formerly often used more loosely; spec. and techn. a young hen from the time she begins to lay till the first moult, after which she is a full-grown hen or fowl.
So it is not really a word for the flesh of the bird itself, but is still used for the livestock. It was first used in the 14th century, right around the same time as poultry began to be attested. Poultry means:
Domestic fowls collectively; those tame birds which are commonly reared for their flesh, eggs, or feathers, and kept in a yard or similar inclosure, as barndoor fowls, ducks, geese, turkeys, guinea-fowls (excluding pigeons, pheasants, etc.); sometimes restricted to the barndoor fowl with its varieties; also applied to the birds as dressed for the market or prepared for food.
Words related to poultry include poulter, poulterer, and poulteress, which are for the people who raise such things, and also poult, which is the young of domestic fowl, or by transference, a child or youth.
(The word poultice is not related, however, and is closer to a pulse or pottage. It is from Latin puls, poltem, from Greek πόλτος. The ou spelling is anomalous, as it is pronounced /ˈpoʊltɪs/.)