Preheat a grill to medium. Prepare the chicken: With your knife parallel to the cutting board, slice each chicken breast in half horizontally without cutting all the way through; open like a book. Cover with plastic wrap and pound until 1/2 inch thick.
Make the filling: Whisk 1/4 cup olive oil, the harissa, cumin and a pinch each of salt and pepper in a bowl. Brush the eggplant with some of the harissa oil. Brush the grill grates with olive oil. Grill the eggplant until tender, 1 to 2 minutes per side. Meanwhile, mix the bulgur, tomato, feta, parsley, mint, lemon juice and remaining 1 tablespoon olive oil in a separate bowl; season with salt and pepper.
Season the pounded chicken on both sides with salt and pepper and brush with half of the remaining harissa oil; top with the eggplant and bulgur mixture, leaving a 1-inch border around the edges. Roll up the chicken toward the pointy end; tie with 3 pieces of kitchen twine. Brush with the remaining harissa oil.
Grill the chicken breasts, turning, until cooked through, 18 to 20 minutes. Transfer to a cutting board and let rest 5 minutes, then remove the twine and slice.