Season the chicken thoroughly with ½ teaspoon of salt, several grinds of black pepper, and paprika.
In a large, heavy-bottomed skillet, heat the oil over medium-high heat. Add the shallots and a pinch of salt, and cook until they are pale golden brown all over, about 5 to 7 minutes.
Add the seasoned chicken to the skillet and sear until it browns lightly on both sides and the shallots darken by a shade or two, approximately 8 to 10 minutes. Transfer the chicken to a plate, leaving the shallots in the pan.
Add the garlic to the pan and cook until fragrant, about 1 minute. Stir in the chopped greens, cilantro, and an additional ¼ teaspoon of salt, tossing to coat them in the residual chicken fat. Although it may appear to be a large quantity of greens, they will reduce in volume as they cook. Cook until the greens just begin to wilt, around 2 minutes. Return the chicken and any accumulated juices from the plate to the skillet with the greens. Pour in enough chicken stock to reach halfway up the sides of the chicken pieces. Bring the liquid to a gentle simmer, partially cover the skillet, and allow the chicken to cook through, about 25 to 35 minutes. Add more stock as needed to keep the greens tender and moist.
Stir in the olives and cook uncovered until they are warmed through, about 1 minute. Squeeze half a lemon over the entire dish, then taste the greens and adjust the seasoning with more salt or lemon juice if desired. Serve the dish topped with flaky salt, a drizzle of extra-virgin olive oil, and an extra sprinkle of smoked paprika.