Look on my works, ye Mighty, and despair!

It’s not immediately obvious how to handle different HTTP methods using Cowboy.
This SO answer, and one of the examples show the way:

handle(Req, State) ->
	{Method, Req2} = cowboy_req:method(Req),
	{ok, Req3} = process_req(Method, Req2),
	{ok, Req3, State}.

process_req(<<"POST">>, Req) ->
	{ok, Body, Req2} = cowboy_req:body_qs(Req),
	_Foo = proplists:get_value(<<"foo">>, Body),
	cowboy_req:reply(200, [
		{<<"content-type">>, <<"text/plain; charset=utf-8">>}
	], <<"ohai">>, Req);

process_req(_, _, Req) ->
	%% Method not allowed.
	cowboy_req:reply(405, Req).