2 tablespoons grated Parmesan cheese, plus more for garnish
4 anchovy fillets, rinsed and minced
1 large head romaine lettuce (1 1/2 pounds), cut into 1-inch pieces
Preheat oven to 375 degrees. Place tortillas on a baking sheet. Brush both sides with 1 teaspoon oil; sprinkle with 1/2 teaspoon chili powder, and season with salt and pepper. Bake until golden brown and crisp, turning once, 8 to 10 minutes. Let cool, then break into pieces.
Heat broiler. In a large bowl, toss shrimp with remaining teaspoon each oil and chili powder. Season with salt and pepper. Lay shrimp flat on a broiler pan, and cook until browned and opaque throughout, turning once, 3 to 4 minutes.
In a small bowl, whisk together mayonnaise, lime juice, Parmesan, anchovies, and up to 2 tablespoons water. Season with salt and pepper.
To serve, toss lettuce with dressing. Divide among plates, and top each with shrimp and broken tortillas. Garnish with more Parmesan.
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