Preheat oven to 400°F (200°C) and position an oven rack in the center of the oven. Line a baking sheet with parchment paper; set aside.
In a food processor or blender, combine the lemon juice and zest, vinegar, anchovy paste, garlic powder, mustard, egg yolk, chili flakes, and three-quarters of the cheese. Process until combined, about 10 seconds. With the processor running, slowly drizzle in 1/2 cup of the olive oil, stopping to scrape down sides as necessary, until mixture is smooth and emulsified, 20 to 30 seconds. Transfer the dressing to a medium bowl.
Whisking constantly, slowly drizzle the remaining 1/4 cup of the olive oil and 3 tablespoons warm water into the dressing. Season with 1/2 teaspoon salt and 1/4 teaspoons pepper. Set aside.
On the parchment-lined baking sheet, make 12 (1 Tablespoon-size) piles of the remaining Parmesan cheese, spacing them 2 inches apart. Bake until the cheese crisps are nicely bubbly and the edges are golden but not browned, 5 to 6 minutes. Let them cool for 3 minutes, then slide them onto a wire rack to cool completely.
To serve, combine the kale and mint in a large bowl. Add the dressing and toss to coat and let sit for 3 minutes. Break the Parmesan crisps into smaller pieces, sprinkle them over the salad, and serve immediately.