Preheat grill to medium-high; grease grate well. Grill burgers for 5 to 6 minutes per side or until internal temperature reaches 160°F (71°C). Meanwhile, heat cast iron or heavy skillet on grill; melt butter. Crack eggs into pan, and season with salt and pepper; close lid and cook for 3 or 4 minutes or until egg whites are firm but yolks are still soft. Gently transfer eggs to plate. Grill pineapple for 2 to 3 minutes per side or until grill-marked. Grill buns for 1 to 2 minutes or until toasted.
Spread mayonnaise over each bun; layer burgers, lettuce, tomatoes, pickled beets, pineapple and fried egg in each bun.
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