Adding custom search button to menu

This code is from the full-screen-search.js file of the plugin in which replace the selector of search input and button with your font awesome button selector as shown below:

jQuery( document ).ready( function( $ ) {

    // ... display the Full Screen search when user clicks the font awesome button
    $( '<font awesome button selector>' ).on( click', function( event ) {
        // Prevent the default action
        event.preventDefault();

        // Display the Full Screen Search
        $( '#full-screen-search' ).addClass( 'open' );

        // Focus on the Full Screen Search Input Field
        $( '#full-screen-search input' ).focus();
    } );

    // Hide the Full Screen search when the user clicks the close button
    $( '#full-screen-search button.close' ).on( 'click', function( event ) {
        // Prevent the default event
        event.preventDefault();

        // Hide the Full Screen Search
        $( '#full-screen-search' ).removeClass( 'open' );
    } );

} );

You can use the above code in custom js file. But if you don’t want to write the repetitive code and if you are ok with editing the plugin js file full-screen-search.js then you can just add your selector as shown below and it will stat showing up overlay when the font awesome icon has focus or it is clicked.

// When the document is ready...
jQuery( document ).ready( function( $ ) {

    // ... display the Full Screen search when:
    // 1. The user focuses on a search field, or
    // 2. The user clicks the Search button
    $( 'form[role=search] input, form[role=search] button, <font awesome button selector>' ).on( 'focus, click', function( event ) {
        // Prevent the default action
        event.preventDefault();

        // Display the Full Screen Search
        $( '#full-screen-search' ).addClass( 'open' );

        // Focus on the Full Screen Search Input Field
        $( '#full-screen-search input' ).focus();
    } );

    // Hide the Full Screen search when the user clicks the close button
    $( '#full-screen-search button.close' ).on( 'click', function( event ) {
        // Prevent the default event
        event.preventDefault();

        // Hide the Full Screen Search
        $( '#full-screen-search' ).removeClass( 'open' );
    } );

} );