In a small saucepan, bring 1 cup water to a boil. Stir in the bulgur and 1/4 teaspoon salt, cover and let stand until the water is absorbed, about 30 minutes.
Meanwhile, in a large bowl, combine the lamb, half of the cilantro, the garlic, paprika and 1 1/2 teaspoons salt. Form into four 1/2-inch-thick patties.
Heat a large nonstick skillet over medium-high heat. Add the patties and cook, turning once, until cooked through, 10 to 12 minutes.
Meanwhile, transfer the bulgur to a large bowl. Stir in the cucumber, grapes, lemon juice, olive oil and remaining cilantro; season with salt and pepper. Serve with the lamb patties.