Heat two tablespoons of olive oil in a large grill pan or skillet. Season chicken breasts with salt and pepper. Place on grill pan and cook until browned on one side (with grill marks, approximately three minutes), then flip to continue cooking on other side. When done, remove from grill and let rest.
Dip cut sides of apricots into honey and place on hot grill pan, cooking until charred and slightly soft.
Toss arugula with remaining olive oil and lemon juice; season with salt and pepper.
Arrange plates with equal portions chicken, apricots, arugula, and blackberries.