Invert a large cast-iron skillet and wrap the bottom with foil; set aside. Toss the chicken with 1 tablespoon olive oil, the garlic, rosemary, red pepper flakes, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Let stand 10 minutes.
Heat 2 tablespoons olive oil in a large nonstick skillet over medium-high heat. Add the chicken and top with the foil-wrapped skillet. Cook until browned on the bottom, 6 to 8 minutes, then flip. Carefully rewrap the skillet with a clean sheet of foil; top the chicken with the skillet. Cook until the chicken is cooked through, about 7 more minutes. Transfer to a cutting board and let rest 5 minutes, then slice.
Meanwhile, pour the wine into the same nonstick skillet and simmer over medium heat until reduced by three-quarters, 1 to 2 minutes. Add the chicken broth and simmer until reduced to about 1/2 cup, about 3 more minutes.
Bring 4 cups salted water to a boil in a medium saucepan. Whisk in the polenta. Cook, whisking, until tender, about 5 minutes. Remove from the heat and stir in the cheese and butter until smooth. Season with salt and pepper.
Toss the arugula with the remaining 1 tablespoon olive oil. Divide the polenta among plates. Top with the chicken, pan sauce and arugula.