Cut the kabocha into bite-sized chunks and trim away most of the skin. Finely chop the onion. Place both in a heatproof bowl together.
Cover the bowl loosely with plastic wrap and microwave at 600W for about 9 minutes, until the kabocha is soft enough to mash easily with a fork.
While the kabocha cooks, cut the processed cheese and ham into small 5 mm cubes.
Add the salt and black pepper to the hot kabocha and mash until no chunks remain. Let it cool slightly so it won't melt the cheese too early.
In a separate bowl, whisk together the egg, cake flour, and water until smooth to make the batter.
Fold the diced ham and cheese cubes into the mashed kabocha until evenly distributed.
Divide the filling and shape into round, slightly flattened patties. Wet or lightly oil your hands if the mixture is sticky.
Dip each patty into the batter, letting the excess drip off, then press firmly into the panko so the whole surface is coated.
Heat oil to 170°C (340°F) and fry the croquettes until deeply golden and crisp on all sides, a few minutes per side. Drain on a wire rack and serve hot while the cheese is still melty.