In a medium-size mixing bowl, combine the flour, ice water, and egg into a thin crepe-like batter. Add the grated cheese, crumbled cooked bacon, cut string beans, and chopped mushrooms and stir to combine.
Heat a nonstick skillet over high heat with the vegetable oil. Using a small ladle, pour the pancake batter into the skillet -- make sure you scoop some of the vegetables into each pancake. Cook until light golden brown, about 2 minutes on each side. Drain on a baking sheet or plate lined with paper towels.
While the pancakes are cooking up, prepare the dipping sauce: Combine the tamari, garlic, lemon juice, sugar, pepper and red pepper flakes in a small bowl. Mix well to combine.
Serve pancakes with the Korean Dipping Sauce on the side.