Will an egg hard boil faster at higher heat? The water in a pot of water will boil at $100°\text{C}$ regardless of how much heat is applied to it after it begins to boil. The higher the heat, the faster the water boils off (but at $100°\text{C}$). The water is getting more thermal energy and dissipates into steam more quickly. Will a fully immersed egg hard boil faster, similarly absorbing more cooking heat energy per second at high heat vs. a low boil? Or, is the hard boiling time only affected by the temperature of the water the egg is immersed in?
 A: Water is a good conductor of heat, but the heat is transferred much more rapidly by convection (i.e. bulk motion of the water).  High heat input will cause very active convection (enhanced by steam bubbles in a "rolling boil"). The rapid motion and turbulance in the water will disturb and thin the boundary layer around the egg surface. The boundary layer is a region of laminar flow where the temperature transitions from the egg surface temperature to the boiling water temperature.  When the boundary layer is disturbed, boiling water will get very close to the egg surface and the heat transfer will be more rapid. 
High heat input does not actually make a big difference to the cooking time because the heat still has to travel through the material of the egg itself.  There are membranes in the egg and anyway the protein coagulates and prevents convection in the egg.
A: A rapid boil will have a higher percentage of the egg's surface area in contact with the higher temperature steam bubbles at any given moment. This will heat the egg slightly faster than a slower or simmering boil.
