Finely chop the zucchini. Lightly salt. Set aside for about 10 minutes. Squeeze out the liquid. Finely chop the carrot, mushrooms, and onion. Combine all the stuffing ingredients well by hand until evenly blended.
Wash both sides of the perilla leaves under running water, one leaf at a time. Shake off the excess water. Do not dry completely. Dredge both sides of the perilla leaves in the flour, one leaf at a time.
Place a perilla leaf on your hand. Add about a tablespoon of stuffing to the upper half of the perilla leaf. Spread evenly with a spoon. Bring the other half of the leaf over to cover the stuffing. Lightly flatten it with your hand. (Do this step for each leaf before the next step.)
Heat a large non-stick skillet and lightly oil, using a couple of teaspoons of oil, over medium low heat. Dip each stuffed leaf, one at a time, in the egg batter, carefully placing it in the heated skillet.
Cook for 2 to 3 minutes each side, adding a teaspoon or two of oil, until slightly golden without browning. Repeat until all the pieces are pan-fried. Serve warm with a dipping sauce.
Alternatively you can stuff zucchinis or peppers:
Cut the zucchini into 1/2 inch-thick rounds. Cut out the interior with a spoon (or a melon baller). Lightly sprinkle with salt. Let sit for about 20 minutes. Pat dry excess water with a paper towel. Dredge in flour. Fill the opening with the stuffing. Coat the stuffed zucchini lightly with the flour. Dip in the egg batter. Follow the directions above to cook.
Cut the peppers in half lengthwise. Dredge in flour. Stuff the peppers. Coat the stuffed peppers lightly with the flour. Dip in the egg batter. Follow the directions above to cook.