In a small saucepan, combine the wine and vinegar and reduce by half over medium heat. Add the chestnuts and cook until they are cooked through and tender, about 10 minutes. Set aside.
Heat a heavy bottom saute pan over medium heat and season the duck on both sides with salt and pepper. Saute the duck breasts slowly skin-side down, draining the fat that accumulates in the pan into a bowl. Reserve the fat.
When the fat has rendered off the duck breasts and the skin is golden brown and crispy, ladle the fat over the tops of the breasts to cook the other side for 2 or 3 minutes, for medium-rare meat. Remove from heat and keep warm.
Drain all but 1 tablespoon of fat from the pan and increase the heat to high. Add the Brussels sprouts and vinegar and cook until they are golden brown and tender. Season, to taste, with salt and pepper.
To serve, pile some Brussels sprouts in the center of a plate. Slice the duck breasts into 5 pieces and lay on top of the greens. Scatter some of the chestnuts about and drizzle the dish with the wine reduction.
Note: To peel the chestnuts, score with an X on the flat side and cook in a pot of boiling water until the outer shell peels back. Drain and remove the peel with a paring knife.