How to Make It
Preheat the oven to 450°. In a small bowl, toss the ham with the mint and parsley. Open the trout on a work surface, skin side down. Rub the flesh with olive oil and season with salt and pepper. Spread the ham mixture on one side of each trout. Close the trout and press to flatten evenly. Secure the trout with 8-inch bamboo skewers. Lightly rub the trout with olive oil and refrigerate them.
In a saucepan, cover the potatoes with water and bring to a boil; boil until tender, about 20 minutes. Drain and let cool, then peel and cut into 1/2-inch dice.
Meanwhile, put the fava beans in a strainer. Lower them into the saucepan of boiling potato water and cook until just tender, about 2 minutes. Transfer the favas to a plate and peel them.
In a medium skillet, heat the 3 tablespoons of olive oil. Add the garlic, onion and red pepper and cook over moderate heat until softened, about 7 minutes. Add the hominy, smoked trout, favas and potatoes and cook, stirring, until heated through, about 2 minutes. Season with salt and pepper. Transfer the mixture to a 10-inch gratin dish and arrange the mozzarella slices on top. Sprinkle with the Parmesan. Lightly drizzle with olive oil and bake in the upper third of the oven for about 10 minutes, or until sizzling and the mozzarella has melted. Leave the oven on.
Meanwhile, heat 3 tablespoons of the vegetable oil in a large skillet. Season the trout with salt and pepper. Add 3 of the trout to the skillet and cook over high heat until browned and crisp, about 4 minutes per side. Transfer the trout to a baking sheet. Discard the oil. Repeat with the remaining 3 tablespoons of vegetable oil and 3 trout. Bake the trout in the oven for about 4 minutes, or until cooked through.
Preheat the broiler. Broil the gratin for about 1 minute, or until richly browned on top. Remove the skewers and transfer the trout to plates. Serve the gratin alongside.