1. Put the garlic, jalapeños, cumin, cardamom, and salt in a food processor. Pulse 3-4 times to get everything well-combined.
2. Coarsely chop the mint leaves and add about a third of them to the food processor along with 4 tablespoons of the olive oil. Pulse a few times. Lift the lid, then scrape the sides of the food processor bowl to incorporate any mint leaves sticking to it, then add the remaining mint leaves and the water. (If you want to sauce to be extra-rich, you could replace the 2 tablespoons of water with olive oil.)
3. Pulse the food processor for a few seconds at a time, stopping to scrape down the sides of the bowl again. Continue to pulse and process, adding additional olive oil, until you reach the desired consistency. The zhoug should be spoonable, the consistency of pesto. Taste, and season with additional salt or spices.