Carte blanche entered the English language as a French loan word in the mid-17th century, when card games were all the rage. A highly fashionable game of the time was piquet, in which a carte blanche was a hand having no face cards. When not playing piquet, English speakers used carte blanche in a literal sense to refer to physical pieces of paper (“carte”) that were blank or white (“blanche”).
By the 18th century, the meaning had expanded to include blank pieces of paper upon which someone signed his name, trusting a second party to come up with the stipulations of a deal. This idea of signing a yet-unwritten contract and handing over authority to the other party led us to the sense most familiar to speakers of modern-day English. Nowadays, if someone has been given carte blanche, it does not mean that she is holding a blank contract or playing cards. It means that she is free to do or say whatever she pleases.
Note that it is a mistake to say “a carte blanche” unless you are talking about a piquet hand or a blank, signed contract. When used in the sense of giving someone free reign, you say they have been given “carte blanche,” and not “a carte blanche.”
On the other hand, blank check, an English term with very similar meanings, is always used with “a” or some other determiner. That term underwent the same progression as carte blanche from its literal meaning to a figurative one (as in Congress gave the president a blank check of unconditional support ). Unlike carte blanche, however, the literal meaning has not fallen out of use. We no longer play piquet, but we still, occasionally, write checks.