For the Beet Chips:
Preheat the oven to 150 degrees F (or to warm). Line 1 baking sheet with paper towels and 2 more pans with silpats or parchment paper.
Trim the end of the beets. Peel the beets, keeping their round shape to make round chips. Slice into paper thin rounds using a knife or mandoline, adjusting the setting if necessary.
In a medium saucepan, combine the vinegar, 1 cup water, and the sugar and bring to a boil, stirring to dissolve the sugar. Add the beets and boil for 1 to 2 minutes to soften slightly and flavor them. Remove from the heat and let the beets stand in the liquid for 5 to 10 minutes, to cool slightly. (The beets can be prepared to this point refrigerated for up to a week in the liquid, before baking.)
Using a slotted spoon, transfer the beets to the towel-lined pan to drain. Lay the beet slices out on the silpat lined baking sheets. The slices can be close to each other, but they should not touch.
Place in the warmed oven to dry for 10 to 12 hours, or until crisp and dry.
Remove the chips from the oven and, using a small offset spatula, transfer them to a cooling rack. Once cooled, the chips can be stored in an airtight container for up to a week.
For the Duck:
Turn the duck breast skin side down. With a sharp knife, make a cut down the length of the center bone, just to break it. Smack the blade across the bones just to break them slightly; do not cut through the meat. Cutting the bones and leaving the breast on them will prevent the meat from shrinking and curling as it cooks. Season the duck breast with salt, pepper, and sugar.
In a large saute pan, heat 1 teaspoon of the olive oil over medium high heat. Reduce the heat to medium. Place the breast in the pan, skin side down, and cook slowly, adjusting the heat if necessary, for about 10 minutes, or until golden brown. When browned, remove the breast from the pan and place on a cutting board. Separate the breast meat from the bone. Chop the carcass into small pieces.
For the Sauce:
Add the chopped bones to the oil remaining in the pan, increase the heat, and saute until the bones are richly browned. Add the shallots and saute for about 2 minutes, or until translucent. Add the garlic, diced beets, 1 teaspoon sugar, and the balsamic vinegar. Pour the red wine into the pan, and bring to a simmer, scraping the bottom to remove all the browned bits. Simmer over low heat for 20 minutes. Season with salt and pepper. Remove from the heat and strain.
Sprinkle the breasts with the cinnamon. Heat the remaining 1 teaspoon olive oil over medium heat in a saute pan just big enough to hold the breast. Place the breasts skin side down in pan, place a small heavy pot on top of the breast to weigh it down, and cook for about 4 minutes to crisp the skin. Remove from the heat.
Just before serving, swirl the butter into the sauce.
Cut the duck breast in half. If serving 4 people, cut each breast half into 2 pieces. Place the duck on the serving plates; spoon the sauce to the side, garnish with a beat chip, and sprinkle the duck with fleur de sel and black pepper. Serve the remaining beet chips on the side, with beaucoup de LOVE.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.