How do I programmatically create a node or entity with an overridden layout?

Layout is constructed from Section and SectionComponent entities, which are then added to the field with the ID of OverridesSectionStorage::FIELD_NAME.

// Create the node, but don't save it just yet.
$node = \Drupal\node\Entity\Node::create([
  'type' => 'page',
  'title' => 'Custom Page',
  'body' => [
    'value' => 'This is the body of the node',
    'format' => 'full_html',
  ],
]);

// Load the field that stores the layout.
$layout_field = $node->get(\Drupal\layout_builder\Plugin\SectionStorage\OverridesSectionStorage::FIELD_NAME);

// Create a new section with a one-column layout.
$section = new \Drupal\layout_builder\Section('layout_onecol');
$region = 'Content';

// Create the configuration for the entity reference to field_banner
// on Nodes of type page.
$banner_configuration = [
  'id' => 'field_block:node:page:field_banner',
  'provider' => 'layout_builder',
  'label_display' => 0,
  'label' => 'Banner',
  'formatter' => [
    'label' => 'hidden',
    'type' => 'entity_reference_entity_view',
    'settings' => [
      'view_mode' => 'full',
    ],
    'third_party_settings' 
  ],
  'context_mapping' => [
    'entity' => 'layout_builder.entity',
    'view_mode' => 'view_mode',
  ],
];

// Create the layout builder component (item/block).
$banner_component = new \Drupal\layout_builder\SectionComponent(\Drupal::service('uuid')->generate(), $region, $banner_configuration);

// Append the component to the section.
$section->appendComponent($banner_component);

// Create the configuration for the body field.
$body_config = [
  'id' => 'field_block:node:page:body',
  'provider' => 'layout_builder',
  'label' => 'Content',
  'formatter' => [
    'label' => 'hidden',
    'type' => 'text_default',
    'settings' => [],
    'third_party_settings' => [],
  ],
  'context_mapping' => [
    'entity' => 'layout_builder.entity',
    'view_mode' => 'view_mode',
  ],
];

// Create a new layout component.
$body_component = new \Drupal\layout_builder\SectionComponent(\Drupal::service('uuid')->generate(), $region, $body_config);

// Append the component to the section.
$section->appendComponent($body_component);

// Set the entire section to the layout field of the node.
$layout_field->appendItem($section);

// Save the node.
$node->save();

To determine the values to use for $banner_configuration and $body_config, I first created a node through the UI, then set it up with the layout builder components I wanted through the UI. Then, I got the NID of the new entity, and dumped it with Drush to determine the values in the layout field:

# 12345 is the Node ID of the node to dump:
drush ev "echo print_r(\Drupal::entityTypeManager()->getStorage('node')->load(12345)->toArray(), TRUE)"

I analyzed the dumped values in the field layout_builder__layout to determine what values to use for the configuration arrays to create the layout component configuration.