Directions

1. In a bowl, combine the ground meats, mixing gently with your hands. Divide the meat into 4 pieces and gently shape each into a 3/4-inch-thick patty. Set the burgers on a baking sheet, cover with plastic wrap, and refrigerate until firm, at least 2 hours.
2. Preheat the oven to 350°F. In a small bowl, coat the beet with 1 teaspoon of the olive oil and sprinkle with a generous pinch of salt. Wrap the beet in foil and roast in a pie plate until tender, 1 hour. Let cool, then peel the beet and slice it 1/4 inch thick. Transfer to a heatproof bowl and add the onion.
3. In a medium saucepan, combine the white vinegar, rice vinegar, sugar, and 2 1/2 tablespoons of salt. Bring to a boil over moderate heat, stirring to dissolve the sugar. Pour the brine over the beet and onion and let cool, about 1 hour.
4. Light a grill or preheat a grill pan. Coat the pineapple slices with the remaining 1 teaspoon of olive oil and a pinch each of salt and crushed red pepper. Grill the pineapple until warm and lightly charred, 2 minutes per side. Transfer to a plate and keep warm.
5. Generously season the burgers on both sides with salt. Cook on grill or grill pan over high heat until browned outside and medium-rare within, about 3 minutes per side. Transfer to a rack and let rest for 3 minutes.
6. In a large nonstick skillet, melt the butter. Add the eggs and cook over moderate heat until the whites are set and the yolks are slightly runny, about 3 minutes. Remove from the heat and season with salt.
7. Set the burgers on the buns, and layer the pickled beets and onions on the burgers. Top with the grilled pineapple rings and the fried eggs and serve.