Not as usual as a double click on a icon, but useful too, you may type a command line for start a browser.
Some brosers have a richfull command line options. If you want to start browsers in a private / incognito mode type:
google-chrome --incognito(for more information check this post from SuperUser about Google-chrome awitches)
(for more information check this article from Mozilla about Command Line Options)
(Epiphany has its own man page, so may type
man epiphany to stay up to date for the options in your installed version).
You may dig into your favourite search engine to no found how many options they offer us.