Remove lamb from refrigerator and let sit at room temperature for 30 minutes before cooking.
In a food processor combine mayonnaise, peppers, garlic, lemon juice 1/2 tsp Salt, 1/4 tsp pepper, and anchovy (if using) until smooth (30 seconds to a minute). Scrape mixture into a bowl cover and chill for at least 30 minutes.
Preheat a grill to high (450°- 550°).
Stir together paprika, cumin, coriander, dry mustard, fennel seed, and the remaining 2 tsp salt and 1 tsp pepper in a small bowl.
Divide lamb into 4 portions; shape each into a patty and brush evenly with oil.
Sprinkle both sides of each patty with paprika mixture. Reserve remaining paprika for future use.
Grill burgers uncovered until a crust forms (about 4 minutes).
Turn burgers and cook until desired doneness, 3-4 minutes for medium.
Top each burger with 2 slices of Manchego cheese. Cover just until cheese melts, about 30 seconds.
Spread cut sides of buns with about 1 Tbsp of the Aioli mixture.
Place 1 burger on each bottom bun and top evenly with Arugula. Cover with top bun and serve.
