In a small bowl, whisk together beet pickling juice, vinegar and mustard. Add oil in a slow stream, whisking until well combined. Season with salt and pepper.
Warm a grill pan over medium-high heat; oil pan. Place apricots on pan, cut-sides down, and cook, turning once, until grill marks form, about 2 minutes. Transfer to a plate.
Divide salad greens, chicken, cheese, almonds, sliced beets and grilled apricots among 4 plates. Drizzle with reserved dressing, season with salt and pepper and serve.