The title of the website (Aviation Humor) should give you a hint. It is a joke!
However, there are a few similar situations, one of which is a pilot requesting a turn to avoid weather. That could go something like this:
Pilot: ABC123 request right turn to avoid
ATC: ABC123 right turn approved, report new heading
Pilot: Approved, WILCO, ABC123
Pilot: ABC123 now heading 120
Though normally, the pilot will give a rough estimate of how much they need to turn ("Request right turn 20 degrees to avoid").
Another example is ATC giving a pilot a relative heading change, and asking the pilot to report the actual heading.
ATC: ABC123 turn right by 10 degrees, report new heading
Pilot: Right by 10 degrees, heading 250, ABC123
The reason ATC doesn't just give the target heading to the pilot directly is that in many cases ATC does not know the heading of a flight - only the track. In cases with strong wind especially, the heading and track may differ significantly.
ATC can also tell a pilot to turn without giving a target heading, during non-gyro operations (compass gyro inoperative, so pilot does not know his heading). But in that case, the pilot will not respond (so you won't be hearing "341, 342, 343..."). ATC will then time the turn, knowing that a standard turn is 3 degrees per second.
ATC: ABC123 make all turns rate 1, start and stop all turns on the word NOW.
ATC: ABC123 turn left NOW.
(wait for a specific number of seconds)
ATC: ABC123 stop turn NOW.