How does compiling circular dependencies work?

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You need to take a 2-pass, or multi-pass approach:

Languages like Java require a multi-pass compiler since the definition of x would not be required to come before the use:

public class Example {  
public static void main(String [] args) {
    assert(x==0);           
    x++;
    assert(x==1);
}
static int x=0;
}

There are various approaches, for example you could do the following:

The first pass could look for all variable declarations, the second for method declarations, etc. until the last pass uses all this information to compile the final code.

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