Warm a large saucepan over medium-high heat, then add oil. When it’s hot, after about 1 minute, add mustard seeds and let them pop, 20 to 30 seconds. Add onion and cook, stirring occasionally, until translucent, 3 to 5 minutes. Add red chiles and cook 20 seconds.
Stir in vinegar, brown sugar, garam masala, ginger, garlic, and cardamom. Bring mixture to a boil over high heat. Mix in peaches and return to a boil; then reduce heat and simmer rapidly, stirring often, until chutney looks jammy and is reduced to about 2 3/4 cups, 50 to 60 minutes.
MAKE AHEAD: Chilled airtight, up to 1 month; serve at room temperature.