Place a large skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Season the chicken breasts with salt and pepper, and sear in the hot skillet until golden brown and cooked through, 4-5 minutes per side.
While the chicken is cooking, place a small skillet over medium-high heat with remaining 2 tablespoons EVOO. Cook the leeks in the pan until tender, 4-5 minutes. Season with salt and pepper, add the chestnuts and about 1/2 cup stock to the pan and bring up to a bubble.