['1/4 cup ketchup', '1/4 cup mayonnaise', '1 teaspoon Asian chile paste such as sambal oelek', '1 1/4 pounds ground beef chuck', '4 kaiser rolls, split', '4 pineapple rings', '1 tablespoon vegetable oil, divided', '4 large eggs', '3/4 cup drained sliced pickled beets']
Prepare grill for direct-heat cooking over medium-hot charcoal (medium heat for gas); see Grilling Procedure .
Combine ketchup, mayonnaise, and chile paste.
Mix beef with 1 teaspoon salt and 1/2 teaspoon pepper, then form into 4 (4 1/4-inch-diameter) patties.
Lightly toast rolls on grill.
Pat pineapple dry and brush with 1/2 tablespoon oil.
Oil grill rack, then grill pineapple and burgers, covered only if using a gas grill, turning once, until pineapple is tender and caramelized and burgers are medium-rare, about 4 minutes total.
Heat remaining 1/2 tablespoon oil in a large nonstick skillet over medium-high heat until hot, then fry eggs.
Spread chile mayonnaise on rolls, then assemble burgers with pineapple, beets, eggs, lettuce, and tomato.
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