For the Chicken: Adjust oven rack to middle position and heat 200º.
Pound each breast between 2 sheets of plastic wrap to a uniform 1/2 inch thickness. Pat the chicken dry with paper towels and season with salt & pepper.
Process the pistachios in a food processor to fine crumbs, about 25 seconds (do not over process). Toss the nuts with the panko, curry powder and cornstarch in a shallow dish. Combine the flour, 1/2 tsp salt and 1/4 tsp pepper in a second shallow dish and whisk the eggs, mustard and garlic in a third shallow dish. Working with 1 chicken breast at a time, dredge in the flour mixture, shaking off the excess, then coat with the egg mixture, allowing the excess to drip off. Finally, coat with the nut mixture, pressing gently so that the crumbs adhere.
Place the nut crusted chicken in a single layer on a wire rack set over a rimmed baking sheet and let sit 5 minutes.
Meanwhile, heat 6 tablespoons of the oil in a 12-inch nonstick skillet over medium heat until shimmering. Add 2 of the chicken breasts and cook until browned on both sides, 4-6 minutes total. Drain the chicken briefly on a paper towel-lined plate then transfer to a wire rack set over a rimmed baking sheet and keep warm in the oven. Repeat with the remaining 6 tablespoons oil and chicken.
For the Chutney: Discard the oil and wipe out the skillet with paper towels. Add the peach preserves, raisins, water, ginger and cayenne to the skillet and simmer over medium-high heat until thick and glossy, 3-5 minutes. Off the heat, stir in the cilantro and vinegar and season to taste with salt & pepper. Serve, passing the chutney separately.