Is there a name for this phenomena that I observed? I wish I had recorded this, but I was about to get some water from my filter as I was thirsty, but I noticed it had frozen in my refrigerator, so I poured some water into it from the refrigerator, and the ice started melting incredibly fast compared to how fast ice normally melts.  It was as if it was some kind of unstable equilibrium.
 A: The thick water provides for a rapid transfer of heat to the ice compared to heat transfer to the ice from ambient air at the same temperature as the water.  This is the same explanation as @AccidentalFourierTransform provided in an earlier comment.
An analogy is why a person in a 40 F lake experiences hypothermia quicker than in 40 degree F air.
You can compare the heat transfer coefficients for the two cases (from water and from air) and see the significant relative difference in the rate of heat transfer to the ice.  Search online for heat transfer coefficients, or see a good textbook on heat transfer; I like an old textbook by McAdams.
A: Quite possibly, the temperature of ice in the refrigerator could have been just near the critical point of water. Water is a complex system. Ice, water and vapour are like three collective states, or arrangements of the system of water molecules, and depending on the parameters like temperature and pressure, you can make the system fall to any of those states.
So, when you gave energy to the system which is already near a phase transition, the system abruptly transitioned to the new equilibrium point.
