In a large skillet, heat 1 tablespoon oil over high; season chicken with salt and pepper. In two batches, cook chicken until opaque throughout, 1 to 2 minutes per side; transfer to a plate. When cool enough to handle, slice chicken crosswise.
In a medium saucepan, bring 2 inches salted water to a boil. Add green beans; cover and cook until crisp-tender, 4 to 6 minutes. Rinse under cold water until cool; drain well.
Make dressing: In a small bowl, whisk together vinegar, jam, mustard, and remaining 2 tablespoons oil; season with salt and pepper.
In a large bowl, toss arugula and radicchio with all but 1 tablespoon dressing. Divide salad among four plates; arrange chicken, green beans, cherries, and almonds on top. Drizzle with remaining dressing; serve immediately.