Place the block of tofu between sheets of paper towels. Set a plate on top; add a 3- to 5-pound weight to the plate. Press tofu for 15 minutes; drain and discard the accumulated liquid.
Meanwhile, combine water, soy sauce, mirin, and dashi in a saucepan. Bring to a boil over medium heat. Remove from heat and set aside.
Remove weight, plate, and paper towels. Cut tofu into 3/4-inch cubes. Place flour in a bowl and lightly coat tofu with flour.
Heat oil in a heavy pot and fry tofu until golden-brown, 4 to 6 minutes. Fry in batches if necessary so as not to overcrowd the pot. Place fried tofu in a serving dish and pour sauce on top. Sprinkle with green onions.