Preheat the oven to 250 degrees F. Whisk the yogurt, mayonnaise, chutney, 2 tablespoons water, 1 teaspoon curry powder, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Add the apple, celery, raisins and cashews and toss; set aside.
Season the turkey cutlets with salt, pepper and the remaining 1 teaspoon curry powder. Heat the olive oil in a large skillet over high heat. Working in batches, cook the cutlets until golden brown and cooked through, about 3 1/2 minutes per side. Transfer to a baking sheet and keep warm in the oven until ready to serve.
Add the lettuce to the bowl with the apple mixture; toss to coat. Serve the turkey cutlets with the salad.