At several restaurants I've been to, in addition to serving an amuse-bouche at the beginning of a meal, there is an equivalent complementary small desert served at the end, sometimes before or after the check.
What on earth is the name for this "course"?
I believe you're thinking of "mignardises." This is what they call it on the french laundry menu, and others.
Another possibility is "Petit Fours", which are small pastries typically served at the end of a meal. I believe there is some overlap in the use of the two terms.
I have never heard a specific name for this that is widely used.
In some places it might be called a a lagniappe (a small free extra, the term originated in Louisiana), or perhaps just a complimentary dessert.
In traditional English formal meals, the final course after dessert would be a savoury — a small, often strongly-flavoured dish such as welsh rarebit or angels on horseback, to ‘clear the palate’ before the port or other digestif.
(As the name indicates, this would not be a sweet dish, so depending what OP means by ‘desert’ [sic], this may not apply.)
If seeking an English language term, I think "Post-Prandial" is a could work well. But I cannot claim that it's ever been used before.
