Both Uses of

immortal
in
A Tale of Two Cities

The form that was to be doomed to be so shamefully mangled, was the sight; the immortal creature that was to be so butchered and torn asunder, yielded the sensation.†
Chpt 2.2 *

Bankruptcy must inevitably have come of this young Pagan, in Lombard-street, London, and also of a curtained alcove in the rear of the immortal boy, and also of a looking-glass let into the wall, and also of clerks not at all old, who danced in public on the slightest provocation.†
Chpt 3.2

Definitions:

(1) (immortal)
living or existing forever

or:

someone famous throughout history

or:

someone who will never die -- such as a mythological god
(2) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

More rarely, "The Immortals" denotes a military corps of the Persian Empire. The Immortals were so-named because each time a member of the 10,000 man corps was killed or seriously wounded, he was replaced by another man. They are best remembered in western culture for their role in defeating the badly out-numbered Spartans at the Battle of Thermopylae.