You never used the word "closure", which makes me think you may not actually get what is happening. In most cases, a lexical is freed (and the reference count decremented if the lexical is a hard ref) at the end of a function call:

sub mock 
{
  my $self = shift; # ref count of the object +1
  # code here 
} # $self goes out of scope, ref count of the object -1
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But since your new One::foo sub is created inside that lexical scope, it prevents $self from being freed (because the created sub "closes" over it, which means it is a closure that keeps the lexical alive for its own use).

sub mock
{
  my $self = shift; # ref count ++
  return sub { $self->{Thing}; };
  # $self is still held by the sub, so it is not freed
}
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So this is the reference that you want to weaken, just add weaken $self; just before *One::foo = sub { $self->{x} = 'x'; return "baz\n"; }; and you will get the expected result.