How to Make It
Trim the sides of the portobello mushrooms by cutting each into a square. Place all the trimmings in a large bowl and cover with 2 1/2 cups of the water. Slice the portobello mushrooms horizontally into paper-thin slices.
Place the soy sauce, ginger, 2 teaspoons of garlic, the jalapeno, cilantro, 1/4 cup of the shallots, and 2 tablespoons of the lemon juice into a mixing bowl. Strain the liquid from the mushroom trimmings and add to the soy sauce mixture. Carefully dip the mushroom slices into the soy mixture and lay the mushroom slices overlapping in a shallow container. Pour the remaining soy sauce mixture over the mushrooms. Cover with plastic wrap and refrigerate for 2 hours.
Line a 4x4 inch pan with plastic wrap. Layer the portobello mushroom slices in an overlapping manner in the pan. Cover the pave with plastic wrap and press down firmly with your hand or a weight. Refrigerate the mushroom pave for 2 hours or overnight. Once ready to serve, invert the pave onto a cutting board. Cut the pave into 4 pieces, and reserve any juices that drip from it.
Slice the white asparagus into 1/4-inch long pieces and toss together with the mint. In a blender, puree the coriander, vinegar, tahini, sesame seeds, grapeseed oil, and the remaining 1 teaspoon of garlic, 2 tablespoons of shallot, 2 tablespoons of lemon juice, and the remaining 1/4 cup of water until smooth. Pass the mixture through a fine-mesh sieve. Fold the asparagus and mint mixture into the vinaigrette and season to taste with salt and pepper.
Place a piece of the pave in the center of each plate. Spoon the reserved juices from the pave around the plate. Drizzle some of the asparagus vinaigrette around the plate and top with freshly ground pepper.