See lines 16 and 42-46 for an example of how to add the name of the month to the Events Grid View

espresso-grid-template.template.php

<?php
// Options
$date_format = get_option( 'date_format' );
$time_format = get_option( 'time_format' );
$temp_month = '';
$reg_button_text = !isset($button_text) ? __('Register Now!', 'event_espresso') : $button_text;
$alt_button_text = !isset($alt_button_text) ? __('View Details', 'event_espresso') : $alt_button_text;//For alternate registration pages

if ( have_posts() ) :
	// allow other stuff
	do_action( 'AHEE__espresso_grid_template_template__before_loop' );
	?>
	<div id="mainwrapper" class="espresso-grid">
	<?php
	// initialize month
	$new_month = '';
	// Start the Loop.
	while ( have_posts() ) : the_post();
		// Include the post TYPE-specific template for the content.
		global $post;

		//Create the event link
		$external_url 		= $post->EE_Event->external_url();
		$button_text		= !empty($external_url) ? $alt_button_text : $reg_button_text;
		$registration_url 	= !empty($external_url) ? $post->EE_Event->external_url() : $post->EE_Event->get_permalink();
		$feature_image_url	= $post->EE_Event->feature_image_url();

		if(!isset($default_image) || $default_image == '') {
			$default_image = EE_GRID_TEMPLATE_URL .'images/default.jpg';
		}

		$image = !empty($feature_image_url) ? $feature_image_url : $default_image;

		$datetimes = EEM_Datetime::instance()->get_datetimes_for_event_ordered_by_start_time( $post->ID, $show_expired, false, 1 );

		$datetime = end( $datetimes );
        if ($datetime instanceof EE_Datetime) {

			$start_date = date_i18n( $date_format, strtotime( $datetime->start_date('Y-m-d') ) );
			$start_time = date_i18n( $time_format, strtotime( $datetime->start_time('H:i:s') ) );
			// add month to markup
			$month = $datetime->start_date('M');
			if ($new_month != $month) {
				echo '<div style="clear:both;"><h3>' .date_i18n('F', strtotime("first day of " . $month)) . '</h3></div>';
				$new_month = $month;
			}
			// end add month to markup
		?>

			<div id="event-id-<?php echo $post->ID; ?>" class="ee_grid_box_v2 item">
				<?php do_action( 'AHEE__espresso_grid_template_template__grid_item_start', $post ); ?>
					<img src="<?php echo $image; ?>" alt="<?php echo sprintf( esc_attr__( '%s Feature Image', 'event_espresso'), $post->post_title ); ?>" />
					<div onclick="" class="darken ee_overlay">
						<p class="event-link"><?php echo '<a class="register-link button" id="a_register_link-' . $post->ID .'" href="' . $registration_url . '">' . $button_text . '</a>'; ?></p>
						<div class="event-title title"><?php echo $post->post_title; ?></div>
						<p class="start-date">
							<span class="event-start-date"><?php echo $start_date; ?></span>
							&nbsp;
							<span class="event-start-time"><?php echo $start_time; ?></span>
						</p>
					</div>
				<?php do_action( 'AHEE__espresso_grid_template_template__grid_item_end', $post ); ?>
			</div>

		<?php
        }
	endwhile;
	echo '</div>';
	// allow moar other stuff
	do_action( 'AHEE__espresso_grid_template_template__after_loop' );

else :
	// If no content, include the "No posts found" template.
	espresso_get_template_part( 'content', 'none' );

endif;