Why do some things on the dinner plate hold heat longer and others less?

Question

For example, the meat may cool pretty quick, but the mashed potatoes are still burning hot.

Answer 1

The thickness to the center is very important to how fast something cools since it affects how much surface there is to the mass. If you have a half pound of butter in a 3" cube, it will have 54 sq inches of surface to lose heat from.  But if you have 8~1 1/2" cubes of butter you will have 81 sq inches of surface to lose heat from.  This is one reason why kids get colder faster than adults do.  The potatoes are thicker than the steak

The outside of the potatoes also dries quickly which stops steam and heat from escaping from it.  The meat stays moist so more stream and heat is lost.

If you stir your potatoes for a minute, much more steam will be lost and you will expose more of the hot part to the air and it will cool down rapidly.

Finally the potatoes have more water than the beef so they can retain more calories of heat than it can.  Water has a higher level of thermal mass per pound(energy stored as heat) than beef does and it must absorb more energy to get it hot than the beef.

Answer 2

It has to do with "specific heat capacity," also known simply as specific heat, which is the measure of the heat energy required to increase the temperature of a unit quantity of a substance by a certain temperature interval. The term originated primarily through the work of Scottish physicist Joseph Black who conducted various heat measurements and used the phrase “capacity for heat.” More heat energy is required to increase the temperature of a substance with high specific heat capacity than one with low specific heat capacity. For instance, eight times the heat energy is required to increase the temperature of an ingot of magnesium as is required for a lead ingot of the same mass. The answer to your question is, the greater the specific heat,  more energy it took to heat it up, and the longer it will take to lose all that heat energy, that is, the longer it will take to cool down.