Remove rosemary leaves from the stems, leaving a few at one end. Set leaves aside. Drop stems into a bowl of water and set aside 15 minutes.
Combine oil, mirin, soy, lemon juice and sugar in a screw-top jar. Season to taste, shake until well combined.
Place mushrooms in a shallow dish and pour over the marinade. Add ¼ cup reserved rosemary leaves and stir to combine. Cover and set aside 15 minutes, turning mushrooms every 5 minutes.
Drain mushrooms, reserving marinade. Drain rosemary stems. Roll prosciutto and thread onto stems alternately with mushrooms.
Preheat a well-greased barbecue plate, barbecue skewers 5-8 minutes, turning and basting with marinade until mushrooms are tender. Serve with mayonnaise or yoghurt.