Why would I run out of water for a few seconds and then get full pressure back
If you have well water, you almost certainly have a pressure tank. The well pump runs until the pressure in the tank (and the overall system) is up to a predesignated level, then shuts off. The tank pressure then pushes water through the system until the pressure drops to a lower predetermined level. Then the pump motor kicks in again. This is supposed to maintain a relatively even pressure at the faucets.

If your pressure tank is not functioning (there is an air valve and an internal separator between water and the air in the tank and the water, either of which can cease to work correctly) this can interfere with maintaining satisfactory pressure and can cause the pump motor to run too frequently. There is also a pressure guage/valve/relay setup, usually a short pipe stem off the main water line near the pressure tank.  If that pipe or valve get clogegd or optherwise stop working properly, the relay doesn't trip at the right time and the pressure can drop very low (even to zero) before pump kicks in again, raising the pressure.

You need someone to determine whether it is the tank or the pressure guage setup that is not functioning and act accordingly.  Tanks are several hundreds of dollars, but a pressure valve fix is much less.