In a large saucepan, add the potatoes and enough salted water to cover by 1 inch. Bring to a boil and cook until almost tender, about 10 minutes. Using a slotted spoon, transfer to a colander. Add the green beans to the boiling water and cook, stirring, for 2 minutes. Drain in the colander with the potatoes; let cool slightly.
In a large bowl, whisk together 1/4 cup olive oil, 1 tablespoon capers, 3 tablespoons parsley, the lemon peel and half of the lemon juice. Add the potatoes and beans and toss; season with salt and pepper.
In a shallow bowl, beat the egg. In another shallow bowl, season the flour with salt and pepper. In a large nonstick skillet, heat the remaining 2 tablespoons olive oil over medium-high heat. Coat the chicken breasts with the seasoned flour, then dip in the egg, allowing any excess to drip off, and coat with the flour again, shaking off the excess. Add to the skillet and cook, turning once, until golden, about 8 minutes. Transfer to a cutting board and cut into strips.
Add the chicken broth to the skillet; cook over medium heat, scraping up any browned bits. Remove from the heat and stir in the remaining 1 tablespoon capers, 1 tablespoon parsley and lemon juice. Top the salad with the chicken and drizzle with the pan sauce.