• 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 tablespoons oil until hot but not smoking. Add the chicken, skin side down, and cook until the skin is well browned, 5 to 6 minutes. Turn and cook until the meat is firm and cooked through, 3 to 5 minutes, depending on thickness. Transfer to a plate. Do not clean the pan.
While the chicken cooks, prepare the remaining ingredients.

• Peel and thinly slice enough shallots to measure ¼ cup.
• Thinly slice the olives crosswise or coarsely chop them.
• If desired, in a fine-mesh strainer, rinse the olives and capers.
• Zest and juice the lemon.
• Strip the parsley leaves from the stems; coarsely chop the leaves for garnish.

In the same pan used for the chicken, if dry, add 1 teaspoon oil and warm over medium-high heat until hot but not smoking. Add the shallots, olives, and capers and season lightly with salt. Cook, stirring occasionally, until the shallots are lightly browned and softened, 2 to 3 minutes.
Stir in the lemon zest and juice, ¼ cup water, and the butter, if using, or 1 tablespoon oil; season to taste with more salt and pepper, if needed. Add the spinach and cook, stirring occasionally, until the spinach is wilted and the sauce has thickened, 2 to 3 minutes.

Transfer the sauce and spinach to individual plates. Top with the chicken, garnish with the parsley, and serve.