Fare-thee-well or fare-you-well are AmE expressions which appear to date back to the late 18th century:
(informal chiefly US) a state of perfection: the steak was cooked to a fare-thee-well. (Collins Dictionary)
According to Etymonline its related meaning, to the last degree is from late 19th century:
Expression to a fare-thee-well "to the last degree" is by 1884, American English.
Its origin is unclear, the Phrase Finder has no clue:
Curiously, the OED has nothing (that I could find) for "fare-thee-well," but has this: "U.S. colloq.
to a fare-you-well: to the last point; to the utmost degree; completely." No explanation in either dictionary of why a synonym for "good-bye" has taken on this meaning.
The AHD appears to suggest that its meaning is an extension of its literary one:
[From fare thee well, may it go well with you, goodbye.]
Where does the current meaning of this AmE idiomatic expression come from?
Is there a reason why it evolved in AmE and not in BrE given that “fare thee well”, literally speaking, were known and used in both sides of the pond?