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.
While the leeks are cooking, toast the bread and spread with about 2 tablespoons butter. Chop the toast into 1-inch cubes and transfer to a large bowl. Add the leeks and chestnuts, parsley and thyme to the bread cubes and toss to combine, cover with foil. Reserve warm.
Place a small pot over medium-high heat with remaining 2 tablespoons of butter. Sprinkle the flour over the melted butter and cook for about a minute. Whisk remaining cup chicken stock, cream and balsamic into the butter-flour mixture, and bring up to a bubble. Simmer until thickened, 2-3 minutes, and season with salt and pepper.
Serve the chicken breasts with the stuffing, balsamic cream gravy and some steamed veggies alongside.