Whisk together the buttermilk, pickle juice, egg, and 1 teaspoon salt in a shallow bowl or baking dish. Add the turkey cutlets and turn to coat. Cover and refrigerate for 1 hour.
In a large shallow bowl, whisk together the flour, cornstarch, garlic powder, dried dill, baking powder, smoked paprika, and remaining salt until evenly combined.
Pour the oil into a large heavy-bottomed pot or Dutch oven and heat to 350°F (175°C). Set up a frying station with a thermometer, a baking sheet fitted with a wire rack, and paper towels nearby.
Remove a turkey cutlet from the marinade and allow any excess liquid to drip off. Dredge it in the flour mixture, return it to the marinade, then dredge it once more in the flour mixture, pressing gently to help the coating adhere. Repeat with the remaining cutlets.
Carefully lower 2 turkey cutlets at a time into the hot oil to avoid overcrowding the pot and dropping the oil temperature. Fry for 4-5 minutes per side, or until golden brown and the internal temperature reaches 165°F (74°C). Transfer to the prepared wire rack and repeat with the remaining cutlets.
Whisk together the Greek yogurt, mayonnaise, fresh dill, lemon juice, and garlic until smooth. Spread the dill yogurt sauce onto the brioche buns. Top the bottom buns with shredded lettuce, crispy turkey cutlets, and bread-and-butter pickle slices. Finish with the top buns and serve immediately.