Preheat a grill to medium high. Coat the chicken legs with 1 tablespoon olive oil, then sprinkle all over with the cinnamon, ginger, cayenne and 3/4 teaspoon salt. Transfer to the grill, skin-side up; cover and cook 12 minutes. Flip the chicken, cover and continue cooking until the skin is golden brown and a thermometer inserted into the thickest part registers 160 degrees F, 10 to 15 more minutes. Grill the grapes, covered but turning occasionally, until slightly charred, about 6 minutes.
Meanwhile, combine 1 1/2 cups water, the butter, lemon zest and a pinch of salt in a medium saucepan; bring to a boil. Add the bulgur, reduce the heat to medium low and simmer, covered, until tender, about 15 minutes. (Drain any excess water, if necessary.)
Fluff the bulgur with a fork and stir in the grilled grapes, discarding the stems. Stir in the scallions, parsley and the remaining 1 tablespoon olive oil; season with salt. Serve with the chicken.