12 ounces baby red potatoes, halved 1 tablespoon olive oil, divided 4 (6-ounce) skinless, boneless chicken breast halves, pounded to 3/4-inch thickness 3/4 teaspoon kosher salt, divided 1/2 teaspoon freshly ground black pepper, divided 2 thyme sprigs 4 ounces cremini mushrooms, quartered 1 tablespoon chopped fresh thyme 1/4 cup whole milk 5 teaspoons all-purpose flour 1 3/4 cups unsalted chicken stock (such as Swanson) 8 very thin lemon slices 1 (8-ounce) package trimmed haricots verts (French green beans) 2 tablespoons chopped fresh flat-leaf parsley