Em filtroDeAuditoria há erro na linha:

Cookie cookie = new Cookies(req.getCookies()).getUsuarioLogado();

as soluções se referem ao construtor e a inexistência de getUsuarioLogado() para Cookies.

Passei o resp como parâmetro para getUsuario.

package br.com.alura.gerenciador.web;

import java.io.IOException;

import javax.servlet.Filter;
import javax.servlet.FilterChain;
import javax.servlet.FilterConfig;
import javax.servlet.ServletException;
import javax.servlet.ServletRequest;
import javax.servlet.ServletResponse;
import javax.servlet.annotation.WebFilter;
import javax.servlet.http.Cookie;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;

import org.apache.tomcat.util.http.Cookies;


@WebFilter(urlPatterns = "/*")
public class filtroDeAuditoria implements Filter{

    @Override
    public void destroy() {
        // TODO Auto-generated method stub

    }

    @Override
    public void doFilter(ServletRequest request, ServletResponse response,
            FilterChain chain) throws IOException, ServletException {
        // TODO Auto-generated method stub
        HttpServletRequest req =(HttpServletRequest) request;
        HttpServletResponse resp = (HttpServletResponse) response;
        String uri = req.getRequestURI();

        String usuario = getUsuario(req, resp);
        System.out.println("Usuário" + usuario + "acessando a uri : " + uri);
        chain.doFilter(request, response);

    }

    private String getUsuario(HttpServletRequest req ,HttpServletResponse resp) {        
        Cookie cookie = new Cookies(req.getCookies()).getUsuarioLogado();
        if(cookie==null) return "<deslogado>";            
        else {
            cookie.setMaxAge(60*10);
            resp.addCookie(cookie);
            return cookie.getValue();
            }
        }


    @Override
    public void init(FilterConfig arg0) throws ServletException {
        // TODO Auto-generated method stub

    }

}