Solenoid valve is the quick and dirty answer.

The right answer is to get your water heater serviced to figure out what’s wrong with it.

It’s not that your water heater is recharging, it’s that when there is no water moving through your pipes, the hot water in the pipe eventually cools off. When you turn on the shower, all the water already in the pipes has to flow out of the shower head before the hot water in the tank heater reaches the showerhead.

You can waste a lot of water constantly running it for ten minutes, but if you’re willing to make the investment you can introduce a hot water recirculating pump into your home. A plumber adjusts your plumbing so that instead of pipes spidering out to all your taps from your hot water tank the hot water lines form a loop that returns back to the hot water heater, and the pump keeps water moving through the loop, so that any give tap always has immediate hot water (or at least warm water, the loop is never perfectly hot so you usually get a warm shower that gets hotter after a couple of minutes).

The downside is that unless your plumbing is easily accessible, the most likely approach to the loop is to use the cold water lines as the return loop, which means that instead of waiting for hot water from a cold shower, you’ll have a tap that is warm out the cold side until the loop clears and then you get cold water. They will usually leave the kitchen sink out of the loop so that people who fill their glass from the tap don’t have to wait.