#!/usr/bin/perl

# https://perlmonks.org/?node_id=1222687

use strict;
use warnings;

my $nobacktick = qr/(?:\\.|[^\\`]++)*+/s;

while( <DATA> )
  {
  s/ $nobacktick \K ` $nobacktick ` / $& =~ s{<}{&lt;}gr /gex;
  print;
  }

__DATA__
This is `my <string>` that I want to modify because the angle bracket is between backticks
This is `my &lt;string>` that I want to modify because the angle bracket is between backticks
This is `a <string> I don't want to modify because it's not between backticks
This is \`another <left_angle_bracket>`I don't want to modify because the first backtick is escaped but I do want to modify <the_last_left_angle_bracket>` between the backticks
`whole string with <string> enclosed in backticks`
`whole string with <string> not enclosed in backticks
`whole string with <string> not enclosed in backticks\`