Pour olive oil into a large frying pan to a depth of 2 inches. Heat over medium until oil reaches 350°F.

In a bowl, combine 1 ½ cups Parmesan, panko, and salt. In a separate bowl, whisk the eggs.

Working in batches, dip zucchini sticks into egg, let excess drip off, then coat in the panko-Parmesan mixture, pressing gently to adhere. Place coated zucchini on a baking sheet.

When oil is hot, fry zucchini in batches until golden brown, about 3 minutes. Transfer to a paper towel–lined plate to drain.

To serve, spread pesto on a platter, arrange zucchini on top, and sprinkle with remaining Parmesan.