Answer: Pan will melt ice faster than the plate
Explanation: Melting ice takes a lot of energy and this energy needs to come from somewhere. Normally this energy is released by cooling down the ice cube's surroundings, which is why you can cool a drink by putting ice cubes into it.
How fast the ice cube melts depends on how fast energy can be transferred from the surroundings into the cube. Materials like metals conduct heat very well, which means that because the cube is cooler than the pan, heat is moved from all over the pan into the cube, so the cube melts quickly. So why did the pan feel cold? As it's a good conductor, it also means that if you put your warm finger on the frying pan heat is conducted out of, and away from, your finger quickly. This makes the pan feel cold.
The plate on the other hand, conducts heat badly, so even though there is quite a large temperature difference between the ice cube and the plate , relatively little heat is transferred. This means that the cube will melt slowly. It also means that heat is not transferred out of your finger quickly, and so the plate feels warm.
Hope this helps :")