Can you refactor the first four lines 
 String rawUrl = t.getRequestURI().getQuery();
 +            HashMap<String, String> paramMap;
 +            String jobId, superstepId, response;
 +            int statusCode;
 +
the try/catch block and the last four lines:

+            ServerUtils.setMandatoryResponseHeaders(t.getResponseHeaders());
 +            t.sendResponseHeaders(statusCode, response.length());
 +            OutputStream os = t.getResponseBody();
 +            os.write(response.getBytes());
 +            os.close();

They look the same in every handler. There should be an abstract class say, DebuggerServerHttpHandler, that extends HttpHandler, which contains the handle method and code you should refactor. And calls an abstract handleFurther or some appropriate name which is then implemented by these handlers.