Clean the squash and cut it in half (don't peel it). Remove the seeds and strings and then chop into roughly 1-inch cubes.
Put 1 gallon of water in a container and set aside. Heat a large heavy-bottom pot over high heat. Add 1 tablespoon of the oil and the squash and sprinkle the brown sugar on top. Cook, stirring occasionally, so the squash pieces mix with the sugar and become caramelized and brown, about 2 minutes. Add the reserved water. Bring to a boil, turn the heat down to medium and let simmer until the squash is tender and slices with no resistance, 8 to 12 minutes.
Meanwhile, heat a frying pan and add the remaining tablespoon of oil. Saute the onions until brown, about 5 minutes. Transfer to a big mixing bowl and set aside.
Drain the squash and return it to the pot. Use a masher to mash the squash; it doesn't have to be perfectly smooth. Add the squash to the onions. Mix together, add 1 teaspoon salt and 1/2 teaspoon pepper and let cool.
Put the breadcrumbs in a flat container. Put the flour in a separate flat container. Crack the eggs in a bowl, add 1 cup of cold water and whisk together.
Scoop the kabocha mixture into roughly 1/2-cup balls. For each ball, roll it in the flour, then in the egg, then cover it with the bread crumbs. When the ball is in the breadcrumb container, press down on it to make a hamburger patty shape. Make sure the breadcrumbs cover the entire korokke (croquette). Put the breaded croquettes aside.
Heat some canola oil to 350 degrees F in a deep frying pan or in a deep-fryer. Toast the hamburger buns. Fry each breaded croquettes until the outside is crispy and golden, about 3 minutes each. Remove and shake lightly to get rid of excess oil.
To build a sandwich, place a lettuce leaf on the bottom of a bun and add a croquette. Top with 4 slices of avocado, some tonkatsu sauce, a tiny pinch of salt and pepper and the top of the bun. Repeat with the remaining ingredients.
To make fresh breadcrumbs, process a pound of fresh or stale bread in a food processor until crumbly (not superfine). Tonkatsu sauce is a Japanese condiment traditionally served with breaded pork cutlets. It is available in Asian grocery stores.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.