Heat a grill to high. In a small saucepan over medium heat, heat the sesame oil. Add the ginger and garlic and cook, stirring, about 2 minutes, until fragrant. Stir in the apricot jam, orange juice, lemon juice, and soy sauce, and bring to a simmer. Remove from heat. Pour out 1/2 cup of the glaze and set aside.
Wrap each shrimp in a slice of prosciutto. Thread the prosciutto-wrapped shrimp and apricot wedges onto 12 skewers (metal, or wooden ones first soaked in water), alternating shrimp and apricots, with 2 of each on each skewer.
Cook on the grill, brushing with the reserved 1/2 cup of the glaze, for about 8 minutes, until grill marks appear. Serve sprinkled with cilantro with extra sauce on the side.