Preheat oven to 450°F. Line a baking sheet with aluminum foil, place an oven-proof baking rack on the sheet, and spray the rack lightly with cooking spray.
Pat the fish filets dry with paper towels. Using a sharp knife, cut the filets across the grain into 1-inch-wide strips.
Set up three shallow bowls for dredging: put the all-purpose flour in the first bowl, lightly whisk the eggs in the second bowl, and combine the panko bread crumbs with 1/2 teaspoon salt and 1/2 teaspoon pepper in the third bowl.
Working with one strip at a time, dredge a piece of fish in the flour and tap off any excess, dip it fully in the whisked eggs, then press it into the panko mixture until it is evenly coated on all sides.
Arrange the coated fish sticks on the prepared rack so they are not touching and there is space for air to circulate.
Bake in the preheated oven for 10 to 15 minutes, flipping each piece once about halfway through baking (about 5–7 minutes in). Bake until the breadcrumbs are golden brown and the fish is opaque and flakes easily with a fork.
While the fish bakes, make the lemon-caper mayonnaise: in a small bowl combine the mayonnaise, drained and finely chopped capers, lemon juice, finely chopped fresh parsley, finely grated lemon zest, and 1/4 teaspoon freshly ground black pepper. Stir until well combined.
When the fish sticks are done, remove them from the oven and let them rest on the rack for 1 minute. Transfer to a serving plate and serve hot with the lemon-caper mayonnaise alongside.