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/lib/recovery-mode/options/dpkg is in friendly-recovery 0.2.25.

This file is owned by root:root, with mode 0o755.

The actual contents of the file can be viewed below.

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#!/bin/sh

. /lib/recovery-mode/l10n.sh

if [ "$1" = "test" ]; then
  echo $(eval_gettext "Repair broken packages")
  exit 0
fi

# do some cleanup
rm /var/lib/apt/lists/partial/*
rm /var/cache/apt/archives/partial/*

# check and use dist-upgraders partial mode if we have it,
# it contains a lot of useful repair code
for v in 2.7 2.6 2.5; do
  python="/usr/bin/python$v"
  script_new="/usr/lib/python$v/dist-packages/DistUpgrade/dist-upgrade.py"
  script_old="/usr/lib/python$v/site-packages/DistUpgrade/dist-upgrade.py"
  if [ -x "$python" ]; then
    if [ -e "$script_new" ]; then
      script=$script_new
    elif [ -e "$script_old" ]; then
      script=$script_old
    else
      continue
    fi
    "$python" "$script" --partial --frontend DistUpgradeViewText \
      --datadir /usr/share/update-manager/
    break
  fi
done

# now run the apt foo again (if u-m was ok that shouldn't be needed
dpkg --configure -a
apt-get update
apt-get install -f
apt-get dist-upgrade

echo ""
echo $(eval_gettext "Finished, please press ENTER")
read TMP

exit 0