Preheat oven to 425°F. Cut chicken breasts in half lengthwise (you'll make a vertical cut, not a horizontal one) and briefly pound each piece with a meat mallet to an even 1-inch thickness. Sprinkle with 1/4 teaspoon each salt and pepper. Place the chicken in a shallow bowl.

Whisk oil, garlic, 1 teaspoon lemon zest, lemon juice and the remaining 3/4 teaspoon salt and 1/4 teaspoon pepper in a small bowl. Pour half of the mixture over the chicken, turning to coat. Set aside to marinate at room temperature while you prepare the vegetables.

Meanwhile, place potatoes, Brussels sprouts, onion and the remaining marinade in a 9-by-13-inch baking dish; toss to coat. Spread the vegetables in an even layer. Sprinkle half the butter pieces over the vegetables. Bake until the vegetables start to brown around the edges, about 25 minutes. Remove from oven, stir the vegetables and place the chicken on top of the vegetables. Pour any leftover marinade into the baking dish. Sprinkle the remaining butter on top of the chicken. Bake until the vegetables are tender and the chicken is cooked through, about 20 more minutes. Sprinkle with thyme and the remaining 1 teaspoon lemon zest.