In The vocabula Review, Richard Lehrer has an article in wich he discusses drugstore drinks"
"In northern New England, they take the term milk shake quite literally. To many residing in that little corner of the country, a milk shake consists of milk mixed with flavored syrup — and nothing more — shaken until foamy. If you live in Rhode Island or in southern Massachusetts and you want ice cream in your milk drink, you ask for a cabinet (named after the square wooden cabinet in which the mixer was encased). If you live farther north, you order a velvet or a frappe (from the French frapper, to ice). "

I have always believed that "frapper" was French meaning to beat. I never had a "frappe" (no accent aigu) that wasn't
beaten, and only ice was icecream.
I am surprised that my dictionary appears to confirm his
description. He also says they are called cabinets in southern New England. I was born about halfway between Boston and Cape Cod, and I never heard of a "cabinet" meaning same as frappe. I wonder why he didn't ;mention "sundae".

frappé
adj.
5Fr, pp. of frapper, to strike6 partly frozen; iced; cooled
n.
1 a dessert made of partly frozen beverages, fruit juices, etc.
2 a drink made of some beverage poured over shaved ice
3 [New England] a milkshake Also, esp. for n. 3, frappe 7frap8