Prepare grill for direct-heat cooking over
medium-hot charcoal (medium heat for gas).
Combine ketchup, mayonnaise, and chile
Mix beef with 1 tsp salt and
½ tsp pepper,
then form into 4 (4¼-inch-diameter)
Lightly toast rolls on grill.
Pat pineapple dry and brush with
½ Tbsp 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 ½
Tbsp 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.