Soak the mushrooms in hot water for 30 minutes; drain. Remove the stems, then dice the caps and place in a food processor. Add the chicken, ham, ginger, 2 teaspoons salt, rice wine, sesame oil, soy sauce, cornstarch and egg white; pulse 3 or 4 times to form a coarse paste. Stir in the scallions and jicama. Cover and chill for 30 minutes.
Cut the top 3 or 4 inches off the cabbage and separate the leaves (reserve the bottom for another use). Place a heaping tablespoon of filling in the center of each leaf and roll up to enclose the filling. Place seam-side down on a plate.
Heat 1/4 inch of oil in a large skillet over medium heat. Add the rolls and fry until the chicken is cooked through and the cabbage is golden, about 2 minutes per side. Drain on paper towels, then serve with soy sauce.