Make the aïoli
Position a rack in the center of the oven and heat the oven to 500°F.
Halve and seed the bell peppers and rub them with oil. Set them cut side down on a heavy-duty rimmed baking sheet and roast until just tender, about 20 minutes. Transfer the peppers to a bowl and cover with plastic wrap to steam, 10 minutes. Pull off and discard the skins. Reserve 4 pepper halves for the farro salad.
Coarsely chop the remaining 2 pepper halves and put in a blender with the mayonnaise, olive oil, garlic, and 2 Tbs. water. Purée until very smooth. Season to taste with salt and pepper. Transfer to a small bowl and set aside.
Make the farro salad
Fill a 4-quart saucepan three-quarters full with water and bring to a boil over high heat. Stir in 2 Tbs. salt and the farro. Cook until tender, 15 to 18 minutes. Drain the farro and spread it on a rimmed baking sheet. Drizzle with 2 Tbs. of the oil and toss to prevent it from sticking.
Cut the reserved roasted peppers into small dice. In an 11-inch straight-sided sauté pan, heat the remaining 2 Tbs. oil over medium heat. Add the onion and cook, stirring with a wooden spoon, until translucent, about 5 minutes. Add the tomatoes, celery, and garlic and cook, stirring occasionally, until softened, 5 minutes more. Add the wine and simmer vigorously until reduced by half, about 12 minutes. Stir in the farro, butter, cheese, oregano, balsamic, and roasted peppers. Season to taste with salt and pepper and more Parmigiano-Reggiano. Remove from the heat, cover, and keep warm.
Cook the chicken
Position a rack in the center of the oven and heat the oven to 200°F.
Put the rice flour in a shallow bowl. Lightly beat the eggs in a second shallow bowl. In a third, combine the panko with the parsley, Parmigiano-Reggiano, 2 tsp. salt, and 1 tsp. pepper.
Butterfly the chicken breasts: Slice the thick part of each breast horizontally almost in half so you can open it like a book. Using a meat mallet, pound each chicken breast between 2 sheets of plastic wrap to an even thickness. Season the chicken with salt and pepper.
Dredge each piece of chicken in the flour, shaking off the excess, then in the egg, letting the excess drip off, and then in the panko mixture to coat well. Transfer to a plate.
Heat 3 Tbs. of the canola oil in a heavy-duty 12-inch skillet over medium heat until shimmering hot—a few crumbs dropped in the oil should sizzle immediately. Cook 2 of the chicken breasts, flipping once with tongs, until golden-brown and just cooked through, 8 to 10 minutes total. Transfer to a plate lined with paper towels to drain briefly, then to a rimmed baking sheet; keep warm in the oven. Repeat with the remaining chicken in 2 batches, cleaning the pan and adding more oil as necessary.
In a medium bowl, toss the arugula with the lemon juice and a pinch of salt, then toss with the olive oil and season with pepper to taste.
Transfer the chicken breasts to a cutting board and cut them into thirds. Serve with the farro, aïoli, and arugula.
nutrition information (per serving):
Photo: Scott Phillips