Preheat oven to 425°F.

Set up a 2-part breading station. Place the whisked eggs in one  part and the panko in the second part.

Working with a small handful of green beans at a time, dip into the eggs and then into the panko, patting to adhere the panko to the vegetables. Place in a single layer, with space between the coated “fries,” on 1-2 jellyroll pans. Repeat with remaining green beans and the asparagus. Using a sweeping motion, spray nonstick cooking spray over the green beans and asparagus. Bake, uncovered, for 12 minutes, until crisp and light golden brown.

In the meantime, prepare the dipping sauce: In a medium bowl, whisk the mayonnaise, mustard, chopped pickle, and pickle juice.

Serve the “fries” with the dipping sauce.