To make the batter: Put the batter ingredients into a medium bowl and whisk until smooth.
Pour the oil into a medium-small saucepan over high heat. When the oil is hot (drop a few drops of batter into it; if it sizzles and bubbles, it's hot) lower the heat to medium-high.
Frying: dip the veggies in the batter, then place them gently in the hot oil. Fry for 1 minute, then, using metal tongs, flip and fry the other side until it’s golden. Remove from the oil and place on a plate lined with a paper towel.
Dipping sauce: put the kombu and mushrooms in a small saucepan with 1 cup of water and bring to a boil. Stir in the soy sauce and mirin (or rice vinegar) and continue to boil for 2-3 minutes. Remove from the heat and let it cool. Serve in a small bowl on the side with the tempura.