1/2 pound feta, crumbled, plus 3/4 cup of the brine
1 tablespoon Louisiana-style hot sauce, preferably Tabasco
Four 4-ounce chicken cutlets
1 tablespoon fresh lemon juice
1/3 cup extra-virgin olive oil, plus more for brushing
Kosher salt
Pepper
Four 6-inch ciabatta rolls, split and toasted
Arugula (not baby), sliced cucumber, red onion and seeded piquillo peppers, for serving
In a large resealable plastic bag, mix the feta brine and hot sauce. Add the chicken cutlets, seal the bag and turn to coat. Refrigerate for 30 minutes.
Meanwhile, in a food processor, pulse the feta cheese and lemon juice. Slowly blend in the 1/3 cup of oil until smooth. Season with salt and pepper.
Preheat a grill pan. Remove the chicken from the brine and pat dry; discard the brine. Brush with oil and season with salt and pepper. Grill the cutlets over moderately high heat, turning once, until cooked through, about 5 minutes total.
Spread the rolls with some of the feta. Top with the chicken, arugula, cucumber, onion and peppers and serve.