Hi Sam,
It's more common in Asia to use fresh coconut to get the coconut cream and milk. We never use can overthere and the size of the fresh coconut is twice bigger than we have here.
Normally when we buy coconut in Asia, the market will break it for us and grate it with special machine (you put the coconut pieces in the middle and push it and the machine will automatically grate it and you'll end up with the grated flesh, around 300 - 400 gr) but if you don't have the machine , you have to properly clean the brown colour on the skin after you break it (so when you cook the coconut milk, you will get a nice clean white colour) & can grate it manually (if you have the patient) and try to grate it horizontally instead vertically so u'll get finer grated flesh and more milk later on. We usually grate it vertically if we want to use the flesh for cake filling, or cook the flesh with spices and mix it with vegetable since it will more texture or rough.
To make coconut cream, pour warm water (3/4 cup) on grated coconut. press it around with your hands and press it through a sieve on top of big bowl & set aside. This one called the coconut cream or thick coconut milk.
After that put more water (1/2 cup) on the same grated coconut, knead & press again through sieve to different bowl until the water finished. Repeat the process depends on the amount needed. (usually u can get around 300 - 400 ml for each).This one called coconut milk or thin coconut milk. It's very different taste and texture from the can one and you can control on how thick and thin you want them to be.
Usually you put the thin coconut milk first when cooking and you have to keep stirring after you put coconut cream in on med heat until it's boiling and keep stirring for a few minutes after that so the coconut won't break up.
If you're not going to use the grated coconut straight away after you grate it, it can be kept in freezer for 1 week&defrost when about to use and apply the same method to get the coconut cream and milk
I have lots of ways to use the coconut cream and milk or the flesh after you obtain the milk ....
for beef, chicken , vegetable, or yellow rice, coconut rice , cake, cookies .. curry either veg, chicken or lamb (indonesian way )
I would recomend one dish call Rendang using braising beef and small potato and you can eat it with plain rice.
Let me know which recipe you want it for and spicy or mild or med ..
i'll send it to you ..
Good luck
Cat