Preheat oven to 450°F.
Place halved potatoes in a medium saucepan; cover with water. Bring to a boil. Reduce heat to maintain a simmer; cook until almost tender, about 8 minutes. Drain.
Meanwhile, heat a large ovenproof skillet over medium-high heat. Add 1 teaspoon oil. Sprinkle 4 chicken breasts with ¼ teaspoon salt and ¼ teaspoon pepper. Add the chicken and 2 thyme sprigs to the pan; cook until browned on the bottom, about 5 minutes. Turn the chicken over. Transfer pan to oven; roast until an instant-read thermometer inserted into the thickest portion of chicken registers 165°F. Transfer the chicken to a plate.
Carefully return the pan to medium-high heat (the handle will be hot). Add the remaining 2 teaspoons oil. Add the parboiled potatoes, cut-sides down, along with the quartered mushrooms and 1 tablespoon thyme; cook, stirring once, until browned, about 3 minutes.
Add 1¾ cups broth, 8 ounces haricots verts and the remaining ½ teaspoon salt; cover and simmer until the beans are just tender, about 3 minutes.
Whisk ¼ cup milk and 5 teaspoons flour in a small bowl; stir into the sauce in the pan. Add the chicken and 5 lemon slices; cover, reduce heat and simmer until the sauce is thickened, about 3 minutes.
Sprinkle with 2 tablespoons parsley.
