“Jones” is an American slang word meaning to be addicted to something, so “jonesing” for something means to crave something very strongly, and generally very vocally.
“Breve” is a coffee drink that is commonly made with half-and-half, which is a product that is equal parts cream and milk. I assume that people have taken to using the term to refer to half-and-half itself, but I’ve not personally heard that.
So the sentence is saying that their cat was addicted to half-and-half and would act like a junkie doing anything to get their next fix.