Wordpress - Pass a PHP variable to another file

The WordPress template functions don't support passing variables (to my knowledge) by default, so you need to write your own function. For example like this,

// functions.php
function my_template_include_with_variables( string $path = '', $data = null ) {
  $file = get_template_directory() . '/' . $path . '.php';
  if ( $path && file_exists( $file ) ) {
    // $data variable is now available in the other file
    include $file;
  }  
}

// in a template file
$critical_css = 'style';
// include header.php and have $critical_css available in the template file
my_template_include_with_variables('header', $critical_css);

// in header.php
var_dump($data); // should contain whatever $critical_css had

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UPDATE 18.8.2020

Since WordPress 5.5 you are able to pass additional arguments with the native template functions (e.g. get_header(), get_template_part(), etc.) to the template files in an array.

From dev docs,

get_template_part( string $slug, string $name = null, array $args = array() )

Simply declaring your variable as global in your header template will suffice:

<?php

global $critical_css;
get_template_part('/path/to/' . $critical_css);

?> 

No need to write your own template loading stuff (which usually is a bad idea).