Combine all the remoulade sauce ingredients together until smooth. Refrigerate until needed.
Prepare 3 bowls - one with the egg, garlic powder, cayenne pepper, and salt & pepper (to taste), another with breadcrumbs, and the third with flour. Start with small amounts of flour and breadcrumbs and add more as needed.
Heat about 1/4" of peanut oil in a heavy pot on medium-high heat. Oil should be kept around 250F (use a deep fry thermometer).
Working quickly, dip each pickle quarter one by one in the flour, followed by the egg, and then the breadcrumbs. Add to the oil and cook for about 4 minutes or until the breading is golden brown. You may need to turn the pickles depending on how large your pot is and how deep the oil is. You will probably have to do a few batches. Place the cooked pickles on a paper towel lined plate to soak up excess oil.
Serve pickles with the remoulade dipping sauce.