This deals with the specific case you present, but I don't think I'm really dealing in a robust way with escaped backticks and escaped escapes. (Aiieee...) And I don't really like capturing stuff then stuffing it back in a replacement. Oh, well... I gotta go to bed now.

c:\@Work\Perl\monks>perl -wMstrict -le
"use 5.010;
 ;;
 use warnings;
 use strict;
 ;;
 use Test::More 'no_plan';
 use Test::NoWarnings;
 ;;
 my @tests = (
   [ 'is `my <string>` that `also <this> one` too',
     'is `my &lgt;string>` that `also &lgt;this> one` too',
     ],
   [ 'is `not <this> one', 'is `not <this> one', ],
   [ 'is \`my <NO> that `but <this> one` yes',
     'is \`my <NO> that `but &lgt;this> one` yes',
     ],
   [ 'is \\\\`my <this> that `but <this> one` no',
     'is \\\\`my &lgt;this> that `but <this> one` no',
     ],
   );
 ;;
 VECTOR:
 for my $ar_vector (@tests) {
   if (not ref $ar_vector) {
     note $ar_vector;
     next VECTOR;
     }
   ;;
   my ($string, $expected) = @$ar_vector;
   ;;
   (my $got = $string) =~ s{
     (?<! (?<! \\) \\) ` [^`]* \K < ([^`]* `)
     }
     {&lgt;$1}xmsg;
   ;;
   is $got, $expected, qq{'$string' -> '$expected'};
   }
 ;;
 done_testing;
"
ok 1 - 'is `my <string>` that `also <this> one` too' -> 'is `my &lgt;s
+tring>` that `also &lgt;this> one` too'
ok 2 - 'is `not <this> one' -> 'is `not <this> one'
ok 3 - 'is \`my <NO> that `but <this> one` yes' -> 'is \`my <NO> that 
+`but &lgt;this> one` yes'
ok 4 - 'is \\`my <this> that `but <this> one` no' -> 'is \\`my &lgt;th
+is> that `but <this> one` no'
1..4
ok 5 - no warnings
1..5
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