Heat oil to 325 degrees. Cut mushrooms crosswise into 3/8-inch-thick slices. Lightly dust them with 2 tablespoons flour; set aside. In a medium bowl, whisk together flour, club soda, eggs, and coarse salt to form a batter.
Dip portobello fries into batter. Blanch fries in hot oil. Drain on a paper towel-lined baking sheet. Let cool. Heat oil to 375 degrees. Cook the blanched fries in hot oil until golden brown and crispy. Drain on a paper towel-lined baking sheet. Season with sea salt, and serve with ponzu sauce.