Preheat oven to 375 degrees. On a baking sheet, brush tortillas on both sides with 1 teaspoon oil. Season with 1/2 teaspoon chili powder, salt, and pepper. Bake until golden brown and crispy, turning once, 10 to 15 minutes. Remove from oven, and let stand until cool. When cool, break into pieces.
Preheat broiler. In a medium bowl, toss shrimp and remaining teaspoons of oil and chili powder. Season with salt and pepper. Arrange shrimp in a single layer on a broiler pan. Broil 4 inches from heat until browned and no longer pink in the middle, 3 to 4 minutes.
In a mixing bowl, whisk together mayonnaise, lime juice, cheese, anchovies, and up to 2 tablespoons water for desired consistency. Season with salt and pepper.
In a large bowl, toss lettuce with as much dressing as desired. Garnish with shrimp, broken tortillas, and cheese.