• If desired, remove the chicken skin and discard (see Make It Leaner note). Pat the chicken dry with a paper towel; season generously with salt and pepper.

In a large frying pan over medium-high heat, warm 2 to 3 teaspoons oil until hot but not smoking. Working in batches if needed, add the chicken and cook, turning once, until slightly browned, 1 to 2 minutes per side. Reduce the heat to medium-low and continue cooking, turning occasionally, until firm and almost cooked through, 3 to 5 minutes. Transfer to a plate. Add more oil between batches if needed. Do not clean the pan.
While the chicken cooks, prepare the remaining ingredients.

• Peel and thinly slice enough onion to measure 1 cup [2 cups].
• Strip the oregano leaves from the stems and coarsely chop enough to measure 2 teaspoons [4 tsp] (the oregano will be divided between the puttanesca sauce and the garnish). Save any remaining oregano for another use.
• Cut the olives in half.
• Rinse the capers.
• Strip the basil leaves from the stems; coarsely chop the leaves for garnish.

In the same pan used for the chicken, stir in half the puttanesca base, or more if desired, and tomato paste and cook over medium-high heat until fragrant and warmed through, 1 to 2 minutes. Add the onion and cook, stirring occasionally, until starting to soften, 2 to 3 minutes. Add the tomatoes, 1 teaspoon [2 tsp] oregano, and ¾ cup [1¼ cups] water, season with salt and pepper, and bring to a boil. Reduce to a simmer and stir in the olives, capers, and kale, in batches if needed.
Nestle the chicken in the sauce and cook until the sauce is thickened slightly and the chicken is cooked through, 2 to 3 minutes. Remove from the heat and season to taste with salt and pepper.

Spoon the sauce into individual bowls and top with the chicken. Garnish with the basil and as much of the remaining chopped oregano as you like and serve.