Pour the canola oil to a depth of 3 inches into a wok or deep, wide saucepan, add the sesame oil (if using), and heat to 350ºF on a deep-frying thermometer or until bubbles immediately form around a wooden chopstick held upright in the pan.
Meanwhile, make the batter. Fill a large bowl with the ice cubes, and then nest a second bowl in the cubes. Add the flour, ice water, egg, and salt to the second bowl and, using a fork or chopsticks, stir lightly just until all the flour is moistened, the egg is incorporated, and the batter is lumpy, rather than smooth. Place near the stove.
To make the dipping sauce, in a small saucepan, heat the dashi over medium-low heat. Add the mirin and soy sauce, stir to mix, remove from the heat, and keep warm. Have the daikon and ginger ready.
Ready the shrimp, onion, pumpkin, sweet potato, and eggplant and place near the stove.
When the oil is ready, using tongs or chopsticks, pick up 1 piece to be fried, dip it briefly into the batter, coating it only lightly, and carefully drop it into the hot oil. Working in batches to avoid crowding, repeat with a few more pieces (this process could take a while, but crowding results in steaming rather than crisping). As the foods fry, use a slotted spoon or wire skimmer to remove bits of fried batter floating in the oil. Fry until the coating is crispy and still light colored but not browned, 3–4 minutes for most foods (for pumpkin and sweet potato, lower the heat slightly so they do not color too much as they fry longer). Using chopsticks or the slotted spoon or skimmer, transfer the pieces to a wire rack or paper towels to drain.
To serve, divide the warm dipping sauce between 2 small bowls and divide the daikon and ginger between 2 small plates for diners to mix to taste into their sauce bowls. Serve the tempura right away.