The prompt_for_user_token function is a shorthand for some functionality provided by Spotipy. Here's the code of the function itself.
https://github.com/plamere/spotipy/blob/master/spotipy/util.py
[ -- snip -- ]
def prompt_for_user_token(username, scope=None, client_id = None,
    client_secret = None, redirect_uri = None, cache_path = None):
''' prompts the user to login if necessary and returns
    the user token suitable for use with the spotipy.Spotify 
    constructor
    Parameters:
     - username - the Spotify username
     - scope - the desired scope of the request
     - client_id - the client id of your app
     - client_secret - the client secret of your app
     - redirect_uri - the redirect URI of your app
     - cache_path - path to location to save tokens
'''
[ -- snip -- ]
cache_path = cache_path or ".cache-" + username
sp_oauth = oauth2.SpotifyOAuth(client_id, client_secret, redirect_uri, 
    scope=scope, cache_path=cache_path)
[ -- snip -- ]
token_info = sp_oauth.get_cached_token()
if not token_info:
    print('''
        User authentication requires interaction with your
        web browser. Once you enter your credentials and
        give authorization, you will be redirected to
        a url.  Paste that url you were directed to to
        complete the authorization.
    ''')
    auth_url = sp_oauth.get_authorize_url()
    try:
        import webbrowser
        webbrowser.open(auth_url)
        print("Opened %s in your browser" % auth_url)
    except:
        print("Please navigate here: %s" % auth_url)
[ -- snip -- ]
You can replicate the code and use auth_url (for example) with popen module and chrome.exe, passing auth_url as an argument.
TIP: Under windows cmd.exe /c start http://your.website/address will forward the address to your current default browser. (as opposed to hardcoding firefox.exe or chrome.exe)