Pour flour onto a plate. Season chicken all over with salt, then coat both sides with flour, letting excess drip off.
In a large skillet over medium-high heat, heat oil. Cook chicken, turning halfway through, until golden brown on both sides, 2 to 3 minutes. Transfer to a plate.
In same skillet over medium-high heat, bring broth, brine, and vodka to a simmer. Cook, stirring occasionally, until slightly thickened, 2 to 3 minutes.
Reduce heat to medium. Add butter, capers, and sliced peperoncini. Cook, stirring frequently, until butter is melted, about 2 minutes.
Nestle chicken back into sauce, coating each side in sauce. Continue to cook until an instant-read thermometer inserted into thickest part of chicken registers 165°, 1 to 2 minutes. Sprinkle with feta.