Combine the broth, wine, and lemon juice in a liquid measuring cup and set aside. Season the thighs on both sides with 2 teaspoons of the salt and all the pepper.
Dump the flour into a gallon-size ziptop freezer bag and add one thigh. Push out as much air as possible and seal the bag. Shake to coat then lay the bag (and the thigh) flat on a countertop or cutting board and pound (as gently as you can) with a meat pounder. When the thigh is about 1/4-inch thick, remove to a plate and repeat with the remaining pieces.
Place a heavy 11-inch, straight-sided sauté pan with a tight-fitting lidA good braise requires a good seal. over high heat for 1 minute. Add 1 tablespoon each of the butter and oil to the pan. When the butter just stops bubbling, carefully slide two thighs into the fat. Reduce the heat to medium-high and cook, giving the pan a gentle shake every now and then, until golden brown — 2 to 4 minutes. Flip the thighs and repeat.
Remove the chicken from the pan to a large plate, add another tablespoon each of butter and oil, and cook another two thighs as before. Follow with the final two thighsAvoid the urge to overcrowd the pan, or the thighs will stew rather than brown..
When the last pieces of chicken exit the pan, immediately add the mushrooms and the remaining teaspoon of salt. Sauté until the mushrooms start to turn golden brown, about 5 minutes.
Stir the mushrooms out to the circumference of the pan and add the capers right to the middle, stirring until they start to brown and pop — 1 minute. Deglaze with a third of the wine-broth mixture, stirring with a wooden spoon to dissolve any browned goodness stuck to the bottom of the pan.
Return the chicken to the pan, overlapping the pieces as needed, and pour in the remaining wine-broth mixture. Arrange the lemon slices atop the chicken, slap on the lid, reduce the heat to low and cook until the chicken is fork-tender — 5 minutes.
Evacuate the thighs and arrange on a serving platter. Boost the heat back to high and sprinkle the remaining teaspoon of flour using a small fine mesh sieve to avoid flour clumps. Cook, whisking constantly for 2 minutes. Add the final tablespoon of butter to the sauce, and continue whisking until it begins to thicken and look glossy, about another minute. Pour the sauce over the chicken, top with the parsley, and serve.