Using a sharp knife (and being careful not to cut yourself), cut a "lid" in the top as you would to carve a jack-o-lantern.
Remove the "lid" and slice off the seedy part, so that you're left with a clean lid or hat; set aside.
Using an ice cream scoop (or large steel spoon), scoop out the seeds and "guts" of the pumpkin. You may have to get in there with your hands, which is always fun (!). Save seeds to roast in the oven later with a little oil and salt, or discard.
Preheat oven to 350°F.
Break the eggs into a large mixing bowl. Using a fork or whisk, beat the eggs by hand (or use an automatic mixer set to low or medium) - about 1 minute.
Add the coconut milk and vanilla. Beat again briefly, until well mixed.
Add the sugar 1/2 cup at a time, stirring as you go, until all sugar is added and has dissolved in the egg mixture. Note: This may seem like too much sugar, but the custard does need to be sweet in order to compensate for the slightly bitter taste of the pumpkin.
Pour or ladle the egg mixture into the pumpkin (try not to spill any on the outside of the pumpkin, or it may burn. If you do spill, wipe it away with a cloth). Don't overfill - leave about 1/2 inch of room room at the top, since the egg mixture will rise a little out of the hole (it is not runny, so this is fine).
Set the pumpkin plus the "lid" together in a baking dish (a long, glass dish works perfectly, like the kind you would use to make lasagna).
Pour about 1/2 inch of water into the bottom of the baking dish (water level should be about a 1/2 inch up the side of the pumpkin).
Being careful not to spill the contents of the pumpkin, place in the oven to bake for 1 hour at 350 degrees. After 1/2 an hour, check the water level, making sure the baking dish doesn't go dry. Tip: Always add warm (or hot) water, never cold. Cold water could make your baking dish shatter.
After 1 hour, you should see the custard rising like a bubble out of the pumpkin. Custard is done when an inserted fork comes out clean. You can also poke the top of the pumpkin with a fork to make sure it is soft enough to eat. If necessary, continue to bake up to 10 minutes more.
Leaving pumpkin in the baking dish, allow to cool, then place in the refrigerator until cold (this will help it "set") Note: The custard may "drop" a little - this is normal.
To serve, replace the "lid" or "hat" of the pumpkin, or set it slightly askew so you can see the custard. Using a sharp serrated knife, cut the pumpkin into pieces (see picture) and place the pieces on serving plates. Eat as is, or serve warm with a little maple syrup or brandy over each piece.