.=method is a mutating call of the method on an object. The call $x.=method does the same as the more verbose assignment $x = $x.method.
In the following example, the $o container initially holds an object of the C class, but after $o.=m(), the value will be replaced with an instance of the D class.
class D { }
class C {
method m() {
return D.new;
}
}
my $o = C.new;
say $o.WHAT; # (C)
$o.=m();
say $o.WHAT; # (D)