Cut tofu into batons about 1/2 in. thick and 3 in. long. Arrange on a non-terry dish towel or triple layer of paper towels set on a plate. Let drain 10 minutes, then blot dry.
Meanwhile, grate ginger with a Microplane, then press through a fine-mesh sieve into a bowl. Measure 1 tsp. juice into a small serving bowl and stir in ketchup.
Heat 1 in. oil in a large wok or wide, deep pot over high heat and insert a deep-fat thermometer. Meanwhile, blot moisture from tofu. When thermometer registers 325°, dredge tofu in flour to coat well. Shake off excess and set on a plate until oil reaches 360°.
Fry tofu 6 or 7 pieces at a time, gently stirring with chopsticks or a slotted spoon so they fry evenly and without sticking, 1 minute, or until crisp (they will still be white; it's fine if they get a little golden, but they'll burst if cooked too long). Return heat to 360° between batches and drain fries on paper towels. If any fries have softened, refry 15 seconds. Sprinkle with salt and serve immediately, with gingered ketchup.