1. Prepare ingredients
Preheat oven to 425 degrees. Rinse all produce. Halve potatoes and place on a baking sheet. Roughly chop parsley. Halve lemon Pat chicken dry with paper towel and place between two large pieces of plastic wrap. Using a meat mallet, pound to 1/4-inch thickness.
2. Roast Potatoes
Toss potatoes on baking sheet with 1 tablespoon olive oil, salt and pepper. Arrange cut-side down in a single layer and roast until brown and crispy, about 20 minutes.
3. Sauté Green Beans
While potatoes roast, heat 1 tablespoon olive oil in large pan over medium-high heat. When oil is shimmering, add green beans and season salt and pepper. Sauté, stirring, until bright green and tender, 5-7 minutes. Transfer green beans to serving plates. Wipe pan clean for the next step.
4. Cook Chicken
Place flour on a large plate and season with salt and pepper. Season chicken all over with salt and pepper. Dredge chicken in flour, shaking off excess. Return pan from green beans to medium-high heat, add butter and 1/2 tablespoon olive oil. When butter is foamy, add chicken; sear until crust is golden and chicken is cooked through, 3-4 minutes per side.
5. Make Piccata Sauce
Transfer chicken to a plate, leaving behind any juices in pan to make sauce. Return pan from chicken to medium-high heat; add capers, white wine, parsley, and juice of 1 lemon. Cook, scraping up browned bits, until liquid is reduced by half, 2-3 minutes. Remove pan from heat and stir in more butter to melt. Return chicken to pan and turn to coat, then taste piccata sauce and add salt and pepper as desired.
6. Plate Chicken Piccata
Divide roasted fingerling potatoes between plates with green beans. Serve with chicken, spooning over piccata sauce from pan. Dig in!