In a blender or food processor, purée the salt, garlic, juniper and allspice until smooth.
Rub the duck with the spice paste, cover and cure overnight in the fridge.
Preheat the oven to 300°F.
Rinse the spice rub off of the duck and place skin side up in a deep baking dish. Cover with duck fat. Roast for 2-3 hours, or until the meat is fall-off-the-bone tender. If the duck is browning too much, cover the pot.
Remove the duck from the fat and set on a wire rack to drain for 30 minutes.
Increase the oven to 400°F.
On a baking sheet, toss the carrots and beets with the oil, salt, pepper and rosemary. Roast for 10-15 minutes, until tender.
On a separate baking sheet, toss the apples with the oil, salt, pepper and cinnamon. Roast for 8-10 minutes, until soft but not mushy.
In a saucepan, bring the chicken stock to a simmer and reduce by 2/3.
In a second small saucepan, heat the sugar over medium-low heat until caramelized, reducing the heat as needed and swirling the pan to avoid burning. Whisk in the red wine vinegar, salt and pepper and bring to a simmer. Simmer until reduced to a syrupy consistency.
Add the vinegar-sugar syrup to the reduced chicken stock, along with the sherry vinegar. Return to a simmer. Season to taste.
To make the purée, add the potatoes and parsnips to a large pot and cover with cold water by 2 inches. Bring to a simmer and cook until soft, about 10-15 minutes. Drain and return to the pot. Add the cream and return to a simmer. Turn off the heat. Use an immersion blender to purée until smooth. Add the butter and whisk until the butter is melted and the purée is light and fluffy, adding more cream if necessary. Season to taste.
Place the duck on a baking dish and return to the oven to crisp up the skin for 5-10 minutes.
Divide the potato-parsnip purée, duck confit and roasted apples between plates. Top with gastrique, pistachios and scallions.