Put the bulgur in a large heatproof bowl. In a small saucepan, bring the broth to a boil. Pour the boiling broth over the bulgur, cover, and let stand until the liquid has been absorbed, about 30 minutes.
In a small nonreactive bowl, whisk together the lemon juice, pomegranate molasses, sugar, 1 1/2 teaspoons salt, and several grinds of pepper until the sugar and salt dissolve. Slowly whisk in the oil to make a dressing. Season to taste with salt and pepper.
In a small bowl, stir together the chickpeas and ½ teaspoon salt. Whisk the dressing to recombine, then add it, along with the chickpeas, to the bowl with the bulgur. Stir to mix well. Cover and refrigerate for 2 hours.
Meanwhile, place a rack in the top third of the oven and preheat the broiler.
Place the bell peppers on a small rimmed baking sheet and broil, turning occasionally, until the skins are charred on all sides, about 10 minutes. Transfer to a bowl, cover, and let steam for 15 minutes. Remove and let cool. Peel and discard the skins, stems, and seeds, and cut the flesh into small dice.
When ready to serve, stir together the pistachios and a pinch of salt in a small bowl. Add the pistachios, roasted peppers, parsley, and dried cherries to the bulgur and toss to mix well. Season to taste with salt and pepper. Divide the salad among bowls and serve right away.