1. Combine bulgur and broth in a medium saucepan and bring to a boil. Cover, reduce heat to low and simmer for 15 minutes. Transfer to a large bowl, fluff with a fork and let stand 15 minutes to cool.
2. Place walnuts in a small nonstick skillet over medium-high heat. Cook, stirring constantly, until toasted and fragrant, about 3 minutes. Transfer to a bowl to cool.
3. Whisk together vinegar, brown sugar, mustard and 1/4 tsp. salt in a small bowl. Add oil in a slow stream and whisk until emulsified. Add apple and chicken to bowl with bulgur. Pour in dressing and toss to coat. Season with salt, transfer salad to a serving bowl or platter and sprinkle with walnuts and scallions. Serve at room temperature.