Vapor pressure is the pressure of a vapor in a thermodynamic equilibrium with its condensed phases in a closed system. When water is in equilibrium (or at boiling point), some of the molecules of water changes into water vapor. Since it is at equilibrium, the reverse of this phase change also happens, where some of the water vapor change back to water as a liquid. The rate of the formation of water vapor depends on the amount of pressure being applied to the closed system. The more pressure being applied, the higher the temperature, thus the more water vapor is produced.
When one is to boil an egg, the person will need sufficient amount of temperature and pressure to boil the egg perfectly. In a situation where one asks to boil the egg at the top of the mountain or near the bottom of the sea, it will be more applicable if the egg were to be boiled near the bottom of the sea. One reason for this is the altitude difference. As one goes up (to the top of the mountain), the temperature decreases continuously, as well as pressure. However, if one goes down (near the bottom of the sea), both temperature and pressure increases.
From here, one can conclude that the atmospheric temperature is lower on top of a mountain compared to a place near the bottom of the sea. If one to boil the egg fast and perfect, they should do it near the bottom of the sea, where pressure and temperature is high. When both of these factors are present and affected, the rate of the reaction changes. So, if the factors were increasing, then the rate of the reaction will increase as well, and thus, the egg will be boiled faster and more perfect near the bottom of the sea.