"""Filters and extracts submodule names under the 'thirdparty/' directory from `git ls-files --stage` output.

This function processes lines from the `git ls-files --stage` command, specifically targeting entries with mode '160000' (indicating submodules). It filters these entries to only include those whose paths start with 'thirdparty/'. The function then extracts the first directory name under 'thirdparty/' and returns a sorted list of unique submodule names.

Args:
    ls_stage_lines (List[str]): A list of strings, each representing a line from the `git ls-files --stage` command output.

Returns:
    List[str]: A sorted list of unique submodule names found under the 'thirdparty/' directory.
"""
You should write code starting with:
from typing import List

def git_submodules_under_thirdparty(ls_stage_lines: List[str]) -> List[str]:
    pass

