Season chicken all over with salt, pepper and paprika.
In a large heavy-bottomed skillet, heat oil over medium-high. Add chicken to the skillet and let sear until it browns lightly on both sides, 8 to 10 minutes.
Transfer chicken to a plate.
Add shallots to the pan with a pinch of salt and cook until pale golden brown all over, 5 to 7 minutes.
Add garlic to the pan and cook until fragrant, 1 minute more.
Add chopped kale, parsley and ¼ teaspoon salt to the pan, stir, tossing to coat in the residual chicken fat. It may seem like a lot of kale so do it in batches until it cooks down. Cook until it has just started to wilt, about 2 minutes.
Add the chicken and any juices on the plate to the skillet with the kale. Pour in the chicken stock. Bring liquid to a gentle simmer. Let everything braise partly covered for 15 minutes. Add more stock as needed to keep the kale tender and moist.
Add the beans. Simmer 5-10 minutes more until the chicken is cooked through and tender.
Stir in olives and simmer 1 minute more.
Squeeze one lemon half over everything. Taste the kale and add more salt or lemon juice if needed.
Serve topped with flaky salt and a drizzle of extra-virgin olive oil.