Place chicken thighs between layers of plastic wrap or inside a plastic zip-top bag (or layers of plastic wrap) on a work surface and pound with meat mallet or rolling pin to 1/4-inch thickness. Season both sides with salt and pepper.
Place flour in a shallow bowl. Dredge each pounded chicken thigh in flour, shaking off any excess flour and transfer to clean plate. (Discard remaining flour.)
Heat 2 tablespoons of the olive oil and 1 tablespoon of the butter in a large skillet over medium-high heat until hot. Add 3 chicken thighs in a single layer and cook until crisp and golden brown, about 4-5 minutes; flip and cook 4-5 minutes more, or until browned and just cooked through. Transfer to serving platter and tent with foil. Repeat with remaining chicken thighs.
Add chicken broth, wine, lemon juice, and capers to same skillet and bring to boil. Cook and stir, scraping up any browned bits, until sauce has thickened, about 3 minutes. Remove from heat, add remaining 2 tablespoons butter and stir to melt; spoon sauce over chicken thighs. Garnish with parsley and lemon slices if using and serve.