Make sure you get the prosciutto sliced at the deli, as it will be thinner and more flavorful than the packaged kind. You'll need 12 metal or soaked wooden skewers (8 to 10 in.).
Prepare a grill for high heat (450° to 550°; you can hold your hand 5 in. above cooking grate only 2 to 4 seconds). Heat oil in a small saucepan over medium heat. Cook garlic and ginger until fragrant, about 2 minutes. Add jam, juices, and soy sauce and cook until simmering. Measure out 1/2 cup glaze and set remaining aside.
Wrap a prosciutto strip around a shrimp and thread onto a skewer. Alternate threading shrimp and apricots, 2 pieces each per skewer.
Cook shrimp skewers, turning as needed and brushing with reserved 1/2 cup glaze, until grill marks appear and shrimp is opaque, about 8 minutes. Sprinkle with green onions and serve with extra sauce on the side.
Note: Nutritional analysis is per serving.