Remove seeds from cardamom pods, and discard pods. Combine cardamom seeds and chile in a spice mill or mortar and pestle; grind into a powder.
Combine cardamom mixture, carrot, and next 5 ingredients in a medium saucepan over medium-high heat. Bring to a boil, and cook, stirring occasionally, 30 minutes. Add vinegar; reduce heat, and simmer 30 more minutes or until thickened. Transfer mixture to a bowl; cover and chill at least 1 hour.