Step 1

Prick duck skin all over with a paring knife or carving fork; season with salt and pepper. Place duck on a large rimmed baking sheet. Toss with thyme, garlic, bay leaves, and juniper berries, pressing aromatics onto legs to adhere. Let sit 30 minutes (or preferably do this the day before; cover and chill).

Step 2

Place racks in upper and lower thirds of oven and preheat to 225°. Heat oil in a large Dutch oven or other heavy pot over medium and cook onion, carrots, and celery, stirring occasionally, until softened, 8–10 minutes. Add wine, bring to a boil, and cook until reduced by half, 6–8 minutes. Add 1 cup water and slip duck legs (including aromatics), skin side down, into liquid. Cover and braise in oven on lower rack until duck is submerged in its fat, 1½–2 hours. Turn duck skin side up and cook, covered, until tender (the bones will wiggle easily in the joint), 1½–2 hours longer.

Step 3

Transfer duck to a deep baking dish; strain liquid into a large measuring glass or medium bowl. Skim fat into dish with duck; set aside braising juices. Chill duck, at least 1 hour and up to 2 days (cover and chill juices if chilling duck more than a couple of hours).

Step 4

While duck is chilling, preheat oven to 225°. Bring milk and 2 cups water to a boil in a large saucepan. Season with salt and pepper; slowly stream in polenta, whisking constantly. Cook, whisking often, until it begins to thicken, about 5 minutes. Cover and transfer to lower rack in oven. Bake until polenta is thick and grains are soft, 20–30 minutes. Whisk to smooth out.

Step 5

While the polenta is in the oven, bring reserved braising juices to a boil in a medium skillet over medium-high heat and cook until thick enough to coat a spoon, 15–20 minutes. Stir in vinegar; keep sauce warm.

Step 6

Remove polenta from oven; keep warm. Increase oven temperature to 400°. Transfer duck legs, leaving fat behind, to a large rimmed baking sheet, placing skin side up. Roast on top rack until skin is crackling crisp, about 20 minutes.

Step 7

Meanwhile, remove ribs and stems from chard leaves by slicing away leaf from both sides of stalk. Slice stalk in half lengthwise; cut into 3"-long pieces. Tear leaves. Heat half of oil in a large skillet over medium-high. Cook garlic, tossing, until golden, about 2 minutes. Transfer to a large bowl. Add stems to skillet and cook, tossing, until crisp-tender, about 5 minutes. Transfer to bowl with garlic.

Step 8

Add remaining oil to skillet; add chard leaves a handful at a time, letting them wilt slightly before adding more, and cook, tossing, until chard is just wilted, about 2 minutes. Season with salt and pepper. Toss in chard stems and garlic and transfer to a platter. Top with red pepper flakes and squeeze lemon over.

Step 9

Whisk Parmesan and butter into polenta. Serve duck with polenta, wilted chard, and sauce alongside.