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Prepare the salsa verde chicken: Pat the chicken thighs dry with a paper towel and season on both sides with salt and pepper. Heat the oil in a large (preferably straight-sided) skillet over medium-high heat. Sear the chicken thighs (in batches, if necessary) skin-side down for 4 to 5 minutes, or until golden brown. Flip the thighs and carefully pour the chicken broth and salsa into the pan. Bring the liquid to a low simmer and braise the chicken for 35 to 40 minutes or until fork tender.
Using tongs, remove the thighs to a large plate or bowl. Bring the liquid in the pan to a low boil. Add the sliced onion and poblano pepper to the pan and cook, stirring every few minutes, for about 8 to 10 minutes, or until the vegetables have softened and the liquid has reduced by half. Using forks, shred the chicken thighs into small pieces, discarding any bones and skin. Reduce the heat to low and add the shredded chicken back into the pan with the vegetables, stirring the mixture together until just combined. Season to taste with salt and pepper, and add the torn cilantro leaves. Keep warm over low heat while you prepare the tortillas and toppings.
Char and warm the tortillas: Line a large plate or bowl with a clean kitchen linen. Turn your gas burner over medium heat. You could use two burners at once to speed this process up, or do them one at a time. Place one tortilla directly over the flame and heat for 15 to 25 seconds per side, flipping them with tongs, or until lightly charred on both sides. Transfer the warm, charred tortillas to the kitchen linen as you work and cover to keep warm.
Assemble: Spoon the chicken mixture into the warm tortillas. Top with a generous sprinkle of feta cheese and several sliced radishes. Finish with cilantro and a dash or two of hot sauce, if using. Serve immediately.