Prepare a fire for grilling, if using coals, or set up a gas grill.
Clean the portabello mushrooms: cut off and discard the dirty stalk bottoms before turning the mushroom upside-down, to prevent loose dirt from falling onto the gills. Then, invert the mushroom (cap down), and remove the remaining stalk by cutting off the base of the cap. Set aside for later use.
With a cleaning brush, remove any loose dirt from the cap, and the underside. For a final cleaning, wipe the cap with a moistened towel. Lay the caps, gills down, on a cutting board. Cut through the caps to make cross section slices about 1/3 inch thick. Pile slices on a plate or tray.
Vigorously shake the marinade in the jar, then pour into a large bowl.
Brush the grill lightly with olive oil.
With tongs, grab 1 1/2 mushrooms worth of mushroom slices, and immerse briefly into the bowl of marinade. Loosen the tongs somewhat and quickly stir/swish the mushrooms around, to let all surfaces of the mushroom get coated. Then immediately pull the slices out, allow to drip-drain for a moment over the bowl, and spread, cut surfaces down, onto the grill.
Note: it is very important to not let the mushrooms sit too long in the marinade?the gills will get soggy, and the end result will be ruined. Grill the mushrooms until well browned on one side. Baste the upper sides with the marinade, and turn. Cook until second side is well grilled. (Total cooking time is 4 to 8 minutes, depending on fire temperature, the mushroom slice-thickness, and desired doneness.)
Remove mushrooms from the grill onto serving plate. Repeat with the remaining mushroom slices. Cool briefly, and enjoy!
Mix all the liquids into a sauce pan, and bring to a high simmer. While heating, coarsely chop the garlic and ginger, and stir into the heating liquid. Cook at a high simmer, uncovered, for a bout 20 minutes, stirring occasionally. Remove from heat, and allow to cool. Pour into a glass jar, seal with a tight lid, and refrigerate overnight. (Marinade can be used immediately, but flavors will improve overnight).