Combine 2 cups water, half of the grated carrots, 2 tablespoons olive oil, the dried apricots, 1/4 teaspoon salt and a few grinds of pepper in a medium saucepan and bring to a boil. Add the couscous; cover and remove from the heat. Set aside until ready to serve.
Meanwhile, mix the yogurt, half of the cilantro, 1 teaspoon paprika and a large pinch of salt in a small bowl; set aside.
Combine the ground beef, scallions, cumin, cinnamon, 1 teaspoon salt and the remaining grated carrot and 1 teaspoon paprika in a large bowl. Mix with your hands until just combined. Form into four 3/4-inch-thick oval patties.
Heat the remaining 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the patties and cook until browned, 4 to 5 minutes per side for medium rare. Fluff the couscous with a fork and stir in the remaining cilantro. Serve the patties with the couscous and yogurt sauce.