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The nicer and the fresher the milk you use, the more delicious your cheese will be. I like to buy my milk the same day I make it into cheese. To warm the milk, you can either get it still warm from the udder (in which case you need to be on a dairy farm) or you can transfer it from the fridge into a large pot and warm it slowly on the stovetop.
One way is to dump acid (vinegar or citric acid) right into the milk to get the correct acidity. This process (called direct acidification) leads to cheeses such as ricotta and mascarpone. The other way to acidify the milk is to add cultures, or living bacteria. Given time, warmth and lack of competitor bacteria, these cultures will eat up the lactose in the milk, turning it into lactic acid.
The most common coagulant is rennet, the name for an enzyme which causes the proteins in milk to link together. Mix the coagulant into the liquid milk and wait until a gel forms.
When you've given the rennet enough time to work on the proteins in the milk, the milk will transform from a liquid into a gel. You can test the 'doneness' of the gel by pressing (with a clean hand) onto the surface of the milk.
The next step is now to cut the curd down from a giant blob into smaller cubes or chunks. You can do this with a 'cheese harp', with a knife or even with a whisk. The size to which you cut the curds will dramatically effect the amount of moisture retained in your final cheese; the smaller the initial pieces, the drier (and more ageable) the cheese will be. And vice versa.
For the next several minutes or even hour (depending on the recipe), you'll stir the curds in the vat. Possibly, you'll turn on the heat and cook the curds while you stir. During this phase, the most important thing that is happening is acid is continuing to develop inside the curd and, from the motion of your stirring, the curds are drying out. The more you cook and the more you stir, the drier your cheese will be.
Finally, it's time to separate the curds from the whey. You might do this nearly final step by simply dumping the contents of the pot into a colander in a sink. You might wait 10 minutes to let the curds settle to the bottom then press the curds together at the bottom of the pot before bringing them up and out of the pot in chunks. Generally, we work quickly at this point in the process because we want to conserve the heat into the curds, encouraging them to mush back together to form a nice smooth wheel. If we wait too long, the curds get cold and the cheese falls apart.
Once the curds have been separated from the whey, you can add salt. Or, you can move the curds into their final forms (or baskets) and press the cheese into a wheel before salting. If a cheese is salted, properly acidified and has the correct amount of moisture inside, it can be aged into something more complex. Or it can be eaten immediately--the same moment it was made.