In the OPed code, the statement
    BEGIN { say __PACKAGE__."::begin $variable"; }
refers to the lexical  $variable scalar, but the  BEGIN block executes first, after the  $variable is declared, but before it is initialized at run time. Hence, it is uninitialized, but still consistent with strict. Changing the variable name makes this clear:

c:\@Work\Perl\monks>perl -wMstrict -le
"use strict;
 use warnings;
 use feature 'say';
 ;;
 Foo->test('xxx');
 ;;
 {
   package Foo;
   ;;
   my $variable = 'bar';
   ;;
   BEGIN { say __PACKAGE__ . qq{::begin $vaRRRRiable}; }
   ;;
   sub test { say $_[0], $_[1], $variable; }
   }
 ;;
 Foo->test('yyy');
"
Global symbol "$vaRRRRiable" requires explicit package name at -e line
+ 1.
BEGIN not safe after errors--compilation aborted at -e line 1.
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Changing the scalar name back to  $variable accesses the lexical again, but it's still not initialized at the time the  BEGIN block runs, nor at the first invocation of  Foo->test('xxx'); (but it is initialized by the time  Foo->test('yyy'); runs):

c:\@Work\Perl\monks>perl -wMstrict -le
"use strict;
 use warnings;
 use feature 'say';
 ;;
 Foo->test('xxx');
 ;;
 {
   package Foo;
   ;;
   my $variable = 'bar';
   ;;
   BEGIN { say __PACKAGE__ . qq{::begin $variable}; }
   ;;
   sub test { say $_[0], $_[1], $variable; }
   }
 ;;
 Foo->test('yyy');
"
Use of uninitialized value $variable in concatenation (.) or string at
+ -e line 1.
Foo::begin
Use of uninitialized value $variable in say at -e line 1.
Fooxxx
Fooyyybar
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