Combine the broth, wine, and lemon juice in a liquid measuring cup and set aside. Season the thighs with 2 teaspoons of the salt and all of the pepper.
Dump the flour into a gallon-size 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 counter 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 and 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 bolden brown — 2 to 4 minutes. Flip the thighs and repeat.
Remove the chicken from the pan to a large plate or platter, add another tablespoon each of butter and oil, and cook another two thighs as before. Then 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 — 5 minutes, give or take.
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 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 the plater. Boost the heat back to high, add the final tablespoon of butter to the sauce, and stir constantly until the sauce begins to thicken — 1 to 2 minutes. Pour over the chicken, garnish with new lemon slices if desired, top with the parsley and serve.