For marinade, whisk buttermilk, mustard, salt and peppers. Pour over cut-up chicken, cover and refrigerate for at least 6 hours and up to a day, stirring occasionally.
For frying, whisk flour with baking powder, salt and pepper in a large flat dish. Lift chicken pieces out of marinade and, without shaking off excess, coat with flour. Let chicken sit in flour, turning occasionally, while heating oil.
Pour oil into a large, heavy-bottomed pot or pan, so that oil only comes halfway up the pot. Heat oil over medium-high until it reaches 350F (180C) (a fryer thermometer is a great tool here). Add 4 pieces of chicken to the pot and reduce heat to medium-low. Fry chicken for 5 minutes, adjusting the heat so it hovers between 280F (145C) and 300F (150C) (oil should be constantly be bubbling) then turn over. Fry 7 minutes more then turn chicken again, frying about another 3 minutes. Transfer chicken to a paper towel-lined baking sheet to drain (and check for doneness by opening chicken or using a probe thermometer).
Return oil heat to 350F (180C) and repeat process with remaining chicken. Cooked chicken can be transferred to an unlined baking sheet and kept warm in a 250F (120C) oven. Serve warm or chill overnight and serve on a picnic!