Preparation

• Heat the oven to 450°F.
• In a glass jar or other lidded container, combine the onions with the lime juice, the sugar, and 1 tsp. salt. Shake well and set aside, shaking occasionally.
• In a medium bowl, combine the turnip, the sweet potato, and 1 Tbs. of the olive oil. Toss well to coat. Arrange the vegetables in a single layer on a large parchment- or foil-lined rimmed baking sheet. Add the beets to the bowl, along with another 1 Tbs. of oil, and toss to coat. Transfer the beets to the baking sheet, keeping them separate from the sweet potato and turnip. Season with salt and pepper. Roast the vegetables, flipping once, until soft on the inside and browned on the outside, 25 to 35 minutes.
• Meanwhile, in a small food chopper or food processor, pulse the garlic, oregano, cumin, 1-1/2 tsp. salt and 1 oz. of the spinach until coarsely chopped. Transfer to a medium bowl, add the lamb and mix well. Shape into 4 patties about 4 inches wide and 1 inch thick.
• Heat a grill pan over medium high heat, and brush with olive oil.  Add the burgers and cook, flipping once, until the burgers are cooked to your liking (145°F for medium).
• Heat a cast-iron or nonstick skillet over medium-high heat. Add the remaining 1 Tbs. of oil, and crack all four eggs into the skillet. Cover and cook until the whites are set, but the yolks are still soft, 3 to 4 minutes.
• Drain the pickled onions. Divide the remaining spinach among the four bottom buns. Pile the pickled onions and roasted vegetable slices on top. Put the burgers on the buns, then top with the fried eggs and serve with the top buns.