Heat 1 teaspoon of the oil in a large saucepan. Add the onion and salt and cook over moderately low heat, stirring, until the onion softens, about 8 minutes. Add the curry and cinnamon stick and cook for 1 minute. Stir in the sugars and vinegar and bring to a boil. Add the mangoes and cook, stirring occasionally, until softened, 35 to 40 minutes. Add the cranberries and cook over moderate heat for 40 minutes, crushing them against the sides of the pan.
Heat the remaining 2 teaspoons of oil in a small skillet. Add the mustard seeds and cook until they begin to pop, then stir them into the chutney. Transfer to a bowl and let cool.
Make Ahead: The chutney can be refrigerated for up to 2 weeks.