Put the potatoes in a medium saucepan and cover with cold water; season with salt. Bring to a boil, then reduce the heat to medium and simmer until almost tender, about 12 minutes. Add the green beans and continue cooking until the beans and potatoes are tender, about 3 more minutes. Drain and transfer to a large bowl. Gently stir in the roasted red peppers, vinegar, scallions, 2 tablespoons olive oil, and salt and pepper to taste; set aside.
Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat a large nonstick skillet over medium-high heat. Add the remaining 1 tablespoon olive oil, then the chicken and cook until golden brown, 2 to 3 minutes per side. Remove from the heat and drizzle with the lemon juice.
Add the arugula to the potato salad and toss; top with the goat cheese and serve with the chicken.
Photograph by Justin Walker
Recipe courtesy of Food Network Magazine