Heat oil in a medium saucepan over medium-high. Add onion, garlic, ginger, and 1 teaspoon salt. Cook, stirring frequently, until onion is golden at edges, 5 to 7 minutes. Stir in tomato paste, cumin, and coriander. Cook 1 minute more.

Add potatoes and broth. Bring to a boil, then add chile, reduce heat, and simmer, partially covered, until potatoes are tender, 15 to 18 minutes. Mash some potatoes against side of pot to thicken. Add turkey; heat through. Remove from heat, stir in yogurt, and season with salt. Serve over rice, with a dollop of yogurt, and mint and sliced chiles.