Prepare quince puree and sauce at least 1 hour ahead of cooking time for duck.
Heat 2 tablespoons of the sugar in a heavy saucepan over medium heat until lightly caramelized. Stir in 1/2 cup of the vinegar and cook over high heat until reduced by half, about 2 minutes. Add diced quince, stock, and salt, and partially cover. Adjust heat to low and cook until fruit is tender, 25 to 30 minutes, stirring occasionally. Puree in an electric blender or with a hand mixer until smooth. Return to the pan and keep warm.
Heat the remaining 1 1/2 tablespoons sugar in a small saucepan over medium heat until lightly caramelized. Add the remaining 3 tablespoons of vinegar and reduce by half. Add quince peelings and demi-glace. Infuse sauce over low heat for 30 minutes, strain, and keep warm.
Heat a heavy skillet until medium hot, over medium-high heat. Season both sides of the duck breasts with salt and pepper. Put duck breast in pan, skin side down, without any butter or fat. Cook for 8 minutes, checking to avoid burning. Remove fat as it accumulates. Flip breasts over, lower heat to medium, and cook 4 minutes longer. Remove duck from heat and keep warm in 1 of the skillets.
To serve, place a small amount of the puree in the center of each of 4 warmed plates. Cut duck breast in half, then cut across the grain into 1/4 inch slices. Shingle the duck slices over the puree. Spoon a small amount of infused quince sauce around the meat. Serve at once.