Like junket, this foodstuff seems to have more or less disappeared from Britain by now (and I’m not sure whether Americans ever had it). It has been displaced by yoghurt and other ready-made puddings/sweets/afters from the supermarket refrigerated display. If we do have it or something like it, we might nowadays call it (up-market) panna cotta or (down-market) shape.
The OED reveals that the word itself has been in English since Chaucer’s day. In the Prologue (1386) he writes
ffor blankmanger [v.r. blankemangere] that made he with the beste
The word does indeed have a French origin, though in modern French it has an extra syllable: blanc-manger blãmãʒe. You can see that it had three syllables for Chaucer, too. Furthermore, at that time it was made from chicken or other meat.
The OED’s first citation for a two-syllable version is 1789, when it was spelt blomange. I wonder when and how it became a sweet dessert and when and how it lost that final syllable.
In EPD Daniel Jones gave the word an alternative pronunciation with a French-style nasalized vowel, -mɔ̃ː(n)ʒ. But that must be based on the French spelling, not on actual French: it’s like connoisseur or epergne, a word that looks French but isn’t.