Weeknight Lemon Chicken Skillet Dinner

Name: Weeknight Lemon Chicken Skillet Dinner
Ingredients: 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; ¾ teaspoon kosher salt, divided; ½ teaspoon freshly ground black pepper, divided; 2 thyme sprigs; 4 ounces cremini mushrooms, quartered; 1 tablespoon chopped fresh thyme; ¼ cup whole milk; 5 teaspoons all-purpose flour; 1 ¾ cups unsalted chicken stock; 8 very thin lemon slices; 1 (8-ounce) package trimmed haricots verts (French green beans); 2 tablespoons chopped fresh flat-leaf parsley