For the duck: Score the skin on the duck breasts in a crosshatch pattern and season on both sides with salt. Place skin-side down in a cold medium sauté pan. Place the pan over low heat and slowly render the fat for 10 minutes, occasionally removing and reserving the fat. Flip the breasts and cook on the flesh side for 2 to 3 minutes. Transfer to a sheet pan, reserving the sauté pan, and bake for 5 minutes. Set aside to rest for 10 minutes and leave the oven on for the sweet potato cake.
Meanwhile, to the reserved sauté pan, add the shallots and salt and turn the heat to medium. If the pan is too dry, add 1 tablespoon of the reserved duck fat. Sweat the shallots, stirring occasionally, until soft and translucent, about 5 minutes. Add 1/4 cup of the stock and deglaze the bits from the bottom of the pan. When the stock is reduced by about half, add the grapes and 1/2 cup of the stock. Bring to a boil, then reduce the stock by about half again.
Remove the pan from the heat and add the port wine. Return to the heat and slightly tilt the pan forward to flambe. Reduce until the alcohol has cooked off, about 2 minutes. Add the remaining 1/4 cup stock and any juices from the duck that have accumulated on the sheet pan and reduce by about half. Finish the sauce with the butter, taste for seasoning and keep warm.
For the sweet potato cake: Coat a small nonstick pan with olive oil. Starting in the center of the pan, arrange a layer of sweet potato slices to cover the bottom of the pan, slightly overlapping them in a circular pattern. Season with salt. Repeat the process with a second layer, adding salt, olive oil and 1 tablespoon of the Parmesan. Repeat this process 4 more times, adding Parmesan on every other layer. On the last layer, add salt and Parmesan, but not olive oil. Firmly press down on the top of the potatoes.
Place the pan over medium-low heat and slowly cook until the bottom of the potatoes is golden brown, 5 to 7 minutes, gently shaking the pan occasionally to prevent the potatoes from sticking. Cover with a flat lid, flip the sweet potato cake onto the lid, then slide the cake back into the pan (the layer of potatoes that was on the bottom will now be on the top). Cook 5 minutes more, then bake until the potatoes are fully cooked through and fork tender, 10 to 15 minutes. Drain off some of the excess fat, then slide the cake onto a cutting board and slice into wedges.
For the haricot verts: While the sweet potato cake bakes, season a pot of boiling water generously with salt. It should be as salty as the sea. Prepare an ice bath and season generously with salt. Add the haricot verts to the boiling water and cook until bright green and tender, but still slightly crisp, 2 to 3 minutes. Remove the haricot verts and immediately plunge into the ice bath.
Coat the bottom of a large sauté pan with olive oil, add the shallots, red pepper flakes and salt and sweat over medium-high heat until the shallots are soft and translucent, 5 minutes. Add a bit of the reserved duck fat and the drained haricot verts and toss to combine. Season with salt to taste, then remove from the heat.
To serve: Slice the duck on the bias, then plate it. Spoon the sauce over the duck and around the plates and garnish with some chives. Serve alongside a wedge of sweet potato cake and the haricot verts.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.