How to Make It
In a sturdy resealable plastic bag, combine the chicken and buttermilk and marinate in the refrigerator for 2 hours.
Preheat the oven to 425°. Place the flour in a large bowl. In a small bowl, stir the 1 teaspoon of salt with the cayenne and mustard powder. Remove the chicken from the marinade, letting the excess drip off, and transfer the chicken to a large plate. Season the chicken all over with the salt mixture.
Set a rack over a baking sheet. In a large cast-iron skillet, heat the shortening until it shimmers. Dredge the chicken in the flour, then fry over moderately high heat, turning, until deep golden, 7 to 8 minutes. Transfer the chicken to the rack and bake for about 25 minutes, until crispy and the juices run clear when a thigh is pierced.
Meanwhile, in a small saucepan of boiling water, blanch the peas for 1 minute. Drain and cool under cold water. Drain well and transfer to a medium bowl. Add the carrots, lemon juice, olive oil and mint. Chop enough carrot tops to make 1/3 cup and add to the bowl. Season with salt and toss to coat.
In the saucepan, bring the vinegar to a simmer. Reduce the heat to low and whisk in the butter, one cube at a time, until incorporated. Whisk in the hot sauce and remove from the heat. Transfer the vegetables to a platter, top with the chicken and serve the buttery hot sauce on the side.