Chicken: Sprinkle chicken with 1/4 tsp each of the salt and pepper. In Dutch oven or large heavy-bottomed saucepan, heat half of the oil over medium-high heat; cook chicken, turning once, until browned, about 6 minutes. Transfer to plate. Set aside.
In same pan, heat half of the remaining oil over medium-high heat; cook apples, turning once, until golden, about 2 minutes. Transfer to separate plate. Set aside.
In same pan, heat remaining oil over medium-high heat; cook onion, garlic and thyme, stirring, until softened, about 2 minutes. Pour in cider; cook, scraping up browned bits, until slightly reduced, about 1 minute.
Stir in broth, mustard and remaining salt and pepper; bring to boil. Add chicken and any accumulated juices. Reduce heat, cover and simmer until juices run clear when chicken is pierced, about 18 minutes.
Transfer chicken to clean plate. Discard thyme. Whisk flour with 1 tbsp water; add to sauce and cook, stirring, until slightly thickened, about 2 minutes. Stir in apples.
Smashed Potatoes: Meanwhile, in large saucepan of boiling salted water, cook potatoes until tender, about 15 minutes; drain and return to pan. Cook over medium heat, stirring, for 1 minute. Add milk, butter, thyme, salt and pepper; coarsely mash together. Serve with apple mixture and chicken.