To make dip (see Recipe Options), place its ingredients in a blender or the cup that came with your immersion (hand) blender. Blend until well combined. Transfer to a serving bowl and cover and refrigerate dip until ready to serve with the avocado fries.
For the avocado fries, line a baking sheet with parchment paper. Place the flour in a wide and sided dish, such as a pie plate. Put the panko in second wide and sided dish and mix in Cajun spice. Place the eggs and lime juice in a medium to large bowl and beat well to combine.
Quarter the avocados lengthwise, pull them apart and discard the pits. Remove the peel from each avocado. Cut each quarter piece of avocado, lengthwise, into three wedges.
Coat a piece of avocado in the flour, shake off excess flour, and then set in the beaten egg, ensuring it’s completely coated. Let the excess egg drip away, and then coat the avocado in the panko, gently pressing it on to help it adhere. Set the coated piece of avocado on the baking sheet. Coat the remaining pieces of avocado in this fashion.
Heat the oil in your deep-fryer to 350˚F. Preheat oven to 200˚F. Set a wire-rack over a clean baking sheet.
Working in batches, deep-fry the avocado fries until richly coloured and crispy, about 2 to 3 minutes. Set cooked avocado fries on the wire rack and keep warm in the oven until the rest are fried. When all are cooked, arrange the avocado fries on a platter with dip and serve.