Heat oil in a large skillet over medium heat. Add onion, 1/4 teaspoon pepper and 1/8 teaspoon salt. Cover and cook, stirring occasionally, until soft and translucent, about 10 minutes. Set aside half the onion for topping; finely chop the other half.
Preheat grill to medium.
Place the chopped onion in a large bowl; add pork, olives , garlic, paprika, the remaining 1/2 teaspoon pepper and 1/8 teaspoon salt. Gently combine, without overmixing, until evenly incorporated. Form into 4 equal patties, about 1/2 inch thick.
Combine mayonnaise, lemon zest, lemon juice and saffron in a small bowl.
Oil the grill rack (see Tip). Grill the burgers, turning once, until an instant-read thermometer inserted in the center registers 165 degrees F , 10 to 12 minutes total. Top with cheese and cook until it is melted, about 1 minute more.
Assemble the burgers on toasted buns with the lemon-saffron mayonnaise, some of the reserved onions and a half a Piquillo (or pimiento) pepper.
Shopping Tip: Spain is known for its superb paprika called Pimenton de la Vera, which has a smoky flavor, and for intensely flavored peppers called Piquillos. Look for these specialty ingredients in well-stocked supermarkets, gourmet- food shops or online at spanishtable.com or tienda.com.
Ingredient Note: Literally the dried stigma from Crocus sativus, saffron is the world's most expensive spice. Each crocus produces only 3 stigma, requiring over 75,000 flowers for each pound of saffron. Fortunately, a little goes a long way. It's used sparingly to add golden yellow color and flavor to a wide variety of Middle Eastern, African and European-inspired foods. Find it in the specialty-herb section of large supermarkets, gourmet-food shops and tienda.com. Wrapped in foil and placed in a container with a tight-fitting lid, it will keep in a cool, dry place for several years.
Tip: To oil the grill rack, oil a folded paper towel, hold it with tongs and rub it over the rack. (Do not use cooking spray on a hot grill.) When grilling delicate foods like tofu and fish, it is helpful to spray the food with cooking spray.
MAKE AHEAD TIP: Cover and refrigerate the lemon-saffron mayonnaise (Step 4) for up to 5 days.