Preheat the oven to 400°F (207°C).
Pat the thawed, cut cod pieces dry with paper towels and place them on a plate.
Spread the 1 cup panko in a thin, even layer on a rimmed baking sheet and bake for about 5 minutes, until lightly browned. Transfer the toasted panko to a shallow dish.
Add the 1/3 cup grated Parmesan and 1/4 tsp salt to the toasted panko, drizzle with the 1 tablespoon vegetable oil, and stir until evenly combined.
Put the 1/3 cup all-purpose flour in a shallow dish and place the 2 large eggs (lightly beaten) in a separate shallow dish.
Set a wire rack on the same rimmed baking sheet and lightly coat the rack with cooking spray.
Working in small batches, coat each fish piece first in flour (shake off excess), then dip it in the beaten eggs (let excess drip), then press it into the panko-Parmesan mixture to coat all sides. Transfer each coated piece to the prepared wire rack, spacing them so air can circulate.
Bake the fish nuggets in the preheated oven until opaque and flaky, about 15 minutes total, turning once halfway through (about 7–8 minutes).
Remove from the oven and serve immediately.