Preheat a grill to medium. Grill 4 large halved and pitted peaches, turning once, until softened and slightly charred, about 8 minutes. Transfer to a bowl, cover and let sit for 10 minutes. Discard the skins. In a food processor, puree the peaches, 1/4 cup dijon mustard and 1 tablespoon each chopped crystallized ginger and sugar; season with salt. Let cool completely before serving.