Step 1

Preheat oven to 350°. Arrange chicken skin in a single layer on a rimmed baking sheet and bake until dark golden brown and crisp all the way through, 12–18 minutes. Let cool, then coarsely chop.

Step 2

Combine chicken skin, garlic, parsley, and lemon zest in a small bowl; season with salt and pepper and toss to combine.

Step 3

Combine lemon, olive oil, and sugar in a small bowl. Season with salt and pepper and let sit at least 10 minutes, mixing once or twice.

Step 4

Do Ahead: Lemon oil can be made 2 days ahead. Cover and chill. Bring to room temperature before using.

Step 5

Season chicken thighs all over with salt and pepper. Cook bacon in a large Dutch oven or other heavy pot over medium-low heat, stirring often, until brown and crisp, 7–10 minutes. Using a slotted spoon, transfer bacon to a small bowl.

Step 6

Increase heat to medium-high. Working in 2 batches if needed, cook chicken thighs, skin side down, in the same pot until skin is golden brown, 7–10 minutes. Transfer to a large plate and turn skin side up. Remove pot from heat. Let chicken cool slightly, then pull skin from meat and transfer to a shallow bowl; cover and chill (save for making the gremolata).

Step 7

Return pot to medium heat and cook shallots and garlic, cut side down, in the same pot, tossing shallots occasionally, until shallots are browned in spots and garlic is golden brown, about 5 minutes. Add wine, stirring to release any bits stuck on the bottom of pot, and cook until reduced by two-thirds, about 2 minutes. Add parsley, bay leaves, reserved bacon, and 8 cups water, season generously with salt and pepper, and bring to a simmer. Return chicken thighs to pot and bring stew back up to a simmer. Cover with a lid, leaving slightly askew so steam can escape, and cook, adjusting heat to maintain a very gentle simmer and skimming foam from surface as needed, until meat is tender and easily pulls away from bones, 1–1½ hours.

Step 8

Transfer thighs to a cutting board and let cool 10–15 minutes; keep liquid at a simmer. Pull meat from bones and tear into bite-size pieces; discard bones. Transfer meat to a medium bowl and cover with plastic wrap. Add kale and simmer until kale is tender and liquid is very flavorful, 25–30 minutes.

Step 9

Remove stew from heat, add meat back to pot, and let cool uncovered until no longer steaming. Cover pot and chill stew at least 12 hours (you can skip this step, but it will dramatically improve the flavor).

Step 10

Uncover stew and spoon off half to three-quarters of fat on the surface; discard. Gently reheat stew until barely simmering. Pluck out and discard parsley, bay leaves, and garlic heads (don't worry about any cloves that may have escaped into stew). Taste and season with more salt and pepper if needed. Ladle into bowls and serve with Chicken Skin Gremolata, Lemon Oil, radishes, and/or red onion as desired.

Step 11

Do Ahead: Stew can be made 3 days ahead. Keep chilled.