It smells like you have a problem with Hydrogen Sulfide in your water. Water absorbs this gas as it passes through the ground. When you turn on your faucet, you actually vent the gas from your tap water.
Hydrogen sulfide can be treated with oxidation and filtration. Oxidation turns the gas into elemental sulfur which can then be removed via filtration. Normally, an aeration system injects air into the water to oxidize the gas. The water then passes through a filter that removes the sulfur. The result is water free of the “rotten egg” smell you describe.