Preheat oven to 425 degrees Fahrenheit.
Set up a two-part breading station. Place the whisked eggs in one part and the panko in the second part.
Working with a small handful of string 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 one to two jelly-roll pans. Repeat with remaining string beans and the asparagus. Using a sweeping motion, spray nonstick cooking spray over the string beans and asparagus. Bake, uncovered, for 12 minutes, until crisp and light golden brown.
Prepare while the veggies are baking.
In a medium bowl, whisk the mayonnaise, mustard, chopped pickle, and pickle juice. Serve alongside the “fries.”