Heat grill on medium. Rub chicken breasts with 1 teaspoon olive oil and season with salt and pepper. In large bowl, toss red plums with 1 tablespoon oil and season with salt and pepper.
Grill chicken until cooked through, 5 to 7 minutes per side. Transfer to cutting board. Grill plums until just charred, 2 to 3 minutes per side; return to bowl and toss with 1 tablespoon oil and scallions.
Slice chicken and add to bowl (along with any juices); toss to combine. Fold in baby arugula, fresh dill, and roasted almonds.