Prepare burgers as recipe directs. Keep patties refrigerated until ready to cook.
Combine olives, capers, garlic and anchovies, if desired, in a food processor. Pulse several times to chop. With blade running, slowly add oil and process until well blended and smooth, stopping to scrape sides of bowl as needed. Makes about 1 1/4 cups.
Melt 1 Tbsp. butter in each of two large nonstick skillets over medium-high heat. Crack four eggs into each skillet. Cover and cook until whites are firm and yolks are slightly runny, about 5 minutes. Remove from heat, uncover and leave eggs in hot skillets.
Grill or broil burgers as recipe directs. Spread 1 Tbsp. of olive mixture over both sides of toasted English muffins or buns. Place burgers on bottoms. Use a spatula to carefully transfer one egg at a time to top each burger. Sprinkle each egg with a pinch of crushed peppercorns. Serve immediately with assorted toppings, if desired.