Cut the butternut squash into large cubes, ~ 1 - 2 inches. Bring a pot of water to a boil, lower to medium heat and boil the squash until tender (not mushy!).
While the squash is cooking, prepare your mushrooms and miso mixture. Roughly slice the mushrooms and sautee in a pan with a dash of sesame oil until tender. Once cool enough to handle, slice into a few smaller pieces and set aside. (see photo below).
Once the squash has cooked, run under cold water to stop the cooking. In a blender, add about ~2/3 c of the squash and a spoonful of water or two, and blend until it is a smooth puree. Set aside. Separately, take 6 - 8 cubes of the cooked squash and slice into a few pieces and set aside. (see photo below).
In a bowl, mix the miso, mirin, soy sauce and 1 c water. Whisk and slowly add 1/2 c of the pureed squash. Set aside.
Add ~ 2 inches of water to a large pot for steaming and bring to a boil. Turn off the heat once boiled while you prepare your custards.
In a large bowl, lightly whisk the eggs to break them up. Slowly add the miso mixture, and lightly whisk together to combine. If you have a fine mesh sieve you can strain the mixture now (will help prevent air bubbles), or just proceed without.
In 4 small ramekins, add a few pieces of sliced mushroom and a few pieces of sliced squash. Then add a quarter of the egg/miso mixture. Do not stir. Bring the pot of water back to a boil and insert a steamer (or makeshift steamer with aluminum foil).
Cover each ramekin with plastic wrap. Place the wrapped ramekins on top of the steamer and place the lid on the pot. Slightly lower the heat to medium high and steam for 10 - 12 minutes or until the custards are set on the edges and jiggling in the middle.
Garnish with more squash, spring onions, and a light drizzle of sesame oil. You can eat them warm or after they have chilled in the refrigerator.