Preheat vegetable oil in a deep fryer or large pot to 350 degrees F.

In a shallow bowl, combine all-purpose flour, salt, and ground white pepper.

Pour buttermilk into another bowl.

Dip each fennel slice in the buttermilk, ensuring it is fully coated.

Dredge the buttermilk-coated fennel in the seasoned flour mixture, ensuring even coverage.

Carefully place the battered fennel slices into the preheated hot oil.

Fry until golden brown and crispy, approximately 2 1/2 minutes.

Remove the fried fennel fries with a slotted spoon and place them on paper towels to drain excess oil.

Lightly season with additional salt to taste.

Prepare the Buttermilk Dipping Sauce: Place the coarsely grated cucumber in a small bowl and sprinkle with 3/4 teaspoon of salt.

Stir to combine and let it sit while preparing the remaining ingredients to draw out excess moisture.

In a separate small bowl, combine buttermilk, mayonnaise, minced garlic, chopped fresh dill, cayenne pepper, and 1/4 teaspoon of salt.

Stir well to combine all ingredients.

Drain the cucumber in a small colander, then use your hands to squeeze out as much excess water as possible.

Add the squeezed cucumber to the buttermilk mixture and stir to thoroughly combine.

Serve the fennel fries immediately with the Buttermilk Dipping Sauce. Alternatively, refrigerate the sauce, covered, for up to 1 day in advance to allow flavors to meld.

Yields approximately 1 1/4 cups of dipping sauce.