Preheat oven to 375 degrees F.

Arrange 2 pieces heavy-duty aluminum foil (or 4 pieces regular aluminum foil doubled up to make 2 pieces) on a work surface.

Place 1 abalone mushroom on each piece of foil. Sprinkle half the sliced garlic and half the sliced fennel on and around each of the mushrooms. Drizzle 1/4 cup of extra-virgin olive oil over each mushroom, being careful not to allow the excess oil to drip off of the foil. Season mushrooms well with salt and pepper and place 1 sprig of rosemary on each mushroom. Seal the foil around the mushrooms, and bake for 20 to 25 minutes, or until tender and fragrant.

While the mushrooms are roasting, combine the crabmeat, 1/2 teaspoon of tarragon, 1/2 teaspoon of chives, and 1/2 teaspoon of parsley in a small bowl. Drizzle the remaining 2 teaspoons of olive oil over the salad and season salt and pepper. Toss well to combine. Chill until ready to serve the mushrooms.

To serve, remove the mushrooms from the foil packages, reserving all the vegetables and juices from each packet. Slice each mushroom thinly on an angle, season again with salt and freshly ground pepper and fan the slices onto 2 appetizer plates. Arrange the sliced vegetables on top of the mushrooms. Sprinkle the crabmeat salad over each mushroom and season with the remaining herbs. Drizzle the reserved oil from the packets over each mushroom and serve immediately.