Crush the cardamom pods with the flat edge of a knife. Pry them open and drop the seeds in a mortar. Put the papery skins in a food processor. Use a pestle to crush the seeds, releasing their fragrance and breaking them down a little.
Pour the seeds in the food processor along with the shells and pulse a couple of times.
Add the mint leaves and pulse until the leaves have broken up and look bruised.
Zest the lemons and add the zest to a food processor along with the sugar. Pulse until the zest, cardamom, mint, and sugar are well incorporated and the sugar is fragrant.
Add the sugar mixture and the water together in a medium saucepan over medium-low heat. Stir until the sugar is completely dissolved.
While the sugar and water are cooking, slice the lemons up. Once the sugar is completely dissolved, add the sliced lemons and the other 1/4 mint leaves (whole mint leaves). Cook for 10 to 15 minutes, or until the lemons have lost their brightness and the liquid is a light yellow color (like lemonade).
Add the vinegar and bring the mixture back to a simmer (about 5 to 8 minutes).
Turn off the heat and add the last 1/4 cup mint leaves. Cover and let sit for 10 minutes.
Strain the lemons, spices, and leaves out using a fine-meshed sieve covered with one layer of cheesecloth.