Preheat a grill to medium. Combine the hoisin sauce, vinegar, 1 tablespoon olive oil, the ginger, 1/2 teaspoon salt and 1/4 teaspoon pepper in a bowl. Remove 3 tablespoons of the mixture to a separate large bowl; add the chicken and toss to coat. Add the plums to the remaining hoisin mixture and toss to coat.
Brush the grill grates with olive oil. Add the chicken, skin-side down, and grill until marked and cooked through, 10 to 12 minutes per side; transfer to a large plate. Place the plums cut-side down on the grill; cook until just softened, 3 to 5 minutes. Transfer to the plate with the chicken.
Toss the arugula with the remaining 1 tablespoon olive oil, the lemon juice and cashews; season with salt. Serve with the grilled chicken and plums.