Put the tofu on a wire rack set over a tray and then place a flat-bottomed plate or tray on top of the tofu. Balance some weights on top of the tray to press the tofu. Let this rest for 1 hour.
For the Karaage marinade, whisk the sake, soy sauce, konbucha, ginger, and garlic together in a bowl and then tear bite-sized pieces of tofu into the marinade. Gently stir this together to ensure the tofu is evenly coated in marinade. Let this marinate for at least 30 minutes.
When the tofu is ready to fry, preheat 1.5 inches of vegetable oil in a heavy-bottomed pot with high sides to 340 degrees F (170 C). Line a wire rack with a few layers of paper towels.
Once the oil is up to temperature, drain the tofu and then roll each piece of tofu in the potato starch to give it an even coating. Dust off any excess starch and set each piece of Karaage into the oil. Flip the tofu periodically to ensure even browning.
When the tofu is light-brown and crisp (about 5 minutes), transfer it to a plate and set it aside. Then, turn up the heat to 355 F (180 C).
Add all of the Karaage back in once the oil is up to temperature and fry it until it is golden brown (another 4-5 minutes).
Transfer the Karaage to the prepared rack and let it drain. Serve immediately with lemon wedges for squeezing.