Summarize:

Use a box cutter or utility knife to cut the cardboard down to size. Aim to have about 10 pieces of cardboard in the end, but it's okay if you have more or less. Place the cardboard pieces in a sanitized 5USgal (19L) bucket. Cardboard egg cartons would also work well for growing the mushrooms. You can sanitize your bucket with a diluted bleach solution. Place a large pot or kettle on the stove until the water boils. Pour the hot water onto the cardboard until it's completely wet through and submerged. This helps kill any organisms that may contaminate your mushrooms later on. Push the cardboard down to compact it if you want to use less water. The boiled water will kill any microorganisms on the cardboard in a process called pasteurization. Put a lid on top of the bucket and let it soak overnight. The heat from the water would kill any mushroom spawn, so it needs to rest until it is cool to the touch. Pour the water out of the bucket and drain the rest of the water from the cardboard squares. Start squeezing on 1 side of the square and work your way across. In the end, they should be damp to the touch, but not completely saturated. Wash your hands with antibacterial soap or use hand sanitizer before handling the cardboard so you don't bring in foreign contaminants. Sprinkle the spawn evenly near the edges of the cardboard square. The king oyster mushrooms prefer to grow from the sides of the cardboard rather than on top of the cardboard. Check your local gardening store to see if they have spawn for king oyster. Otherwise, king oyster mushroom spawn can be purchased online. Continue building layers of the cardboard and spawn until you're out of cardboard. Don't place any of the spawn on top of the last piece of cardboard since it will not be covered and may not grow as effectively. You'll need about 100 mushroom spawn total to grow on 10 layers of cardboard, but you don't have to be precise with your count.
Cut cardboard into 4 by 4in (10 by 10cm) squares and put them in a bucket. Boil water and pour it over the cardboard until it is completely saturated. Cover the bucket and let it cool for 8 hours. Squeeze the cardboard to drain any excess water. Place 10 mushroom spawn on 1 piece of cardboard. Stack the squares on top of one another, placing spawn on each layer.