Mitt, dear chap, one is delighted to escort Muffy to the cotillion.
type of dance, 1766, from French cotillion (15c.), originally "petticoat," a double diminutive of Old French cote "skirt" (see coat (n.)); its application to a kind of dance arose in France and is considered obscure by some linguists, but there are lively turns in the dance that flash the petticoats.
Meaning "formal ball" is 1898, American English, short for cotillion ball. French uses -on (from Latin -onem) to reinforce Latin nouns felt to need more emphatic power (e.g. poisson from Latin piscis). It also uses -on to form diminutives, often strengthened by the insertion of -ill-, as in the case of this word.