Where do I hook to have the server do something in PHP on block attribute change?

I’m not sure that I understood exactly your goal, but check if hooking to render_block does what you want, as it will modify the block’s output before it gets rendered in the frontend. Usage example:

add_filter( 'render_block', 'my_dynamic_data', 10, 2 );

function my_dynamic_data( $block_content, $block ) {

    if('my_value' === $block['attrs']['myAttribute'] ) {
       // Do something
    }   

    return $block_content;
}