Preheat the oven to 425 degrees F (220 degrees C). Set 2 shelves near the center of the oven. Spray a baking pan with cooking spray; place frozen chicken filets on the pan.

Bake in the preheated oven on the lower-center shelf until hot and crisp, about 20 minutes, turning filets over halfway through heating time. When cool enough to handle, cut filets into bite-sized pieces.

Meanwhile, line a large rimmed baking sheet with parchment paper, and add tater tots in an even layer.

Bake on upper-center shelf in the preheated oven until tots are soft and slightly baked, 8 to 10 minutes. Remove from oven. Spray cooking spray on the bottom of a cup. Smash each tater tot into a flat disk with cup.

Return to the oven; bake until golden brown and crispy, 15 to 20 minutes. Remove from oven again, and sprinkle with 1/2 cup Parmesan cheese.

Bake again until cheese is melted, about 1 minute. Set aside to cool, about 5 minutes.

Meanwhile, whisk Caesar dressing, lemon juice, anchovy paste, garlic, 1/4 teaspoon black pepper, and salt together in a bowl until smooth.

Toss together lettuce, chicken, remaining 1/2 cup Parmesan cheese and remaining 1/4 teaspoon black pepper until well combined. Pour half the dressing into salad mix and toss until evenly coated. Add cooled tater tots and toss until well combined.

Transfer to a serving platter, drizzle with white balsamic vinegar, and garnish with more Parmesan cheese. Serve immediately.