Restless Shadelings begin to spawn at exactly midnight PST, in The Master's Cellar near the meeting stone for Karazhan. There are two entrances to the cellars, which are right across from each other. They continue to spawn for most of the morning until about 9:00 AM PST, when they despawn. They do not spawn with adds, so you fight only the Restless Shadeling in the pet battle.
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Capturing a Restless Shadeling is a requirement for the Battle Pet achievement: