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