Heat 2 tablespoons oil in a large oven-safe skillet over medium-high heat. Pat the chicken dry and season both sides with salt and pepper. Lay the breasts smooth-side down in the skillet and cook, turning once, until golden on both sides, about 8 minutes total. Transfer the skillet to the oven and bake until just cooked through, 10 to 12 minutes more.
Heat the remaining 1 tablespoon oil in a large skillet over medium-high heat. Add the green beans and cook about 2 minutes. Season with salt and pepper, add 1/4 cup water then cover and cook until the beans are tender, about 3 more minutes.
Meanwhile, mix the butter, tarragon, lemon juice and shallot in a small bowl until well combined; season with salt and pepper.
Top each chicken breast with a dollop of the butter mixture and serve with the green beans and lemon wedges.
If fresh tarragon isn't available, feel free to use 1 teaspoon of dried tarragon.
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