Have you tried setting the output of the first do_shortcode inside the second call?
$output = do_shortcode('[first_shortcode]Some content[/first_shortcode]');
echo do_shortcode('[second_shortcode]'.$output.'[/second_shortcode]');
Have you tried setting the output of the first do_shortcode inside the second call?
$output = do_shortcode('[first_shortcode]Some content[/first_shortcode]');
echo do_shortcode('[second_shortcode]'.$output.'[/second_shortcode]');