Why would heating left-overs in a ring heat them more evenly?
I believe the unspecified context of this question is cooking in an oven or microwave.  No matter what your cooking method, the heat is introduced at the outside of the food:


Microwaves only penetrate a short distance; the rest of the cooking through to the center is by conduction (or convection, if the food is liquid)
Conduction/convection in an oven only interact with the surface of the food
Radiant heat (infrared essentially), one of the main cooking modes in an oven, again only affects the surface


By creating a ring shape, if it is practical, you create more surface for the heating modality to interact with the food.

This is the reason for an angel food cake pan or bundt pan having a tube at the center--it allows more surface area to be involved in the cooking process.  

Also, with the metal pan, the metal has a higher conductivity of heat than does air.  This is why parts of the cake in contact with the pan (or a non-insulated metal cookie sheet, or whatever) generally brown more than the parts exposed to the air.

However, in practice except for things cooked in tube pans, I wouldn't really worry about this affect.  If you can stir occasionally, it becomes close to moot.

Note that in pan frying or sauteing or or griddling or stir frying (depending on the term used in context), the main surface area that is relevant is the bottom of the food, that which is contact with the pan.  A ring doesn't help much here, compared to spreading the food out.

I looked at the first part of that thread.  It does not seem to be a hot bed of science and knowledge of cookery.