Index: src/wp-admin/includes/class-walker-nav-menu-edit.php
===================================================================
--- src/wp-admin/includes/class-walker-nav-menu-edit.php	(revision 35323)
+++ src/wp-admin/includes/class-walker-nav-menu-edit.php	(working copy)
@@ -232,7 +232,23 @@
 			</div><!-- .menu-item-settings-->
 			<ul class="menu-item-transport"></ul>
 		<?php
-		$output .= ob_get_clean();
+		$item_output = ob_get_clean();
+
+		/**
+		 * Filter a menu item's starting output.
+		 *
+		 * The menu item's starting output only includes `$args->before`, the opening `<a>`,
+		 * the menu item's title, the closing `</a>`, and `$args->after`. Currently, there is
+		 * no filter for modifying the opening and closing `<li>` for a menu item.
+		 *
+		 * @since 4.4.0
+		 *
+		 * @param string $item_output The menu item's starting HTML output.
+		 * @param object $item        Menu item data object.
+		 * @param int    $depth       Depth of menu item. Used for padding.
+		 * @param array  $args        An array of {@see wp_nav_menu()} arguments.
+		 */
+		$output .= apply_filters( 'walker_nav_menu_edit_start_el', $item_output, $item, $depth, $args );
 	}
 
 } // Walker_Nav_Menu_Edit