I haven't tried it, but you might be able to setup two client instances like below, or something similar.
char serverName1[] = "web.server1.net";
char serverName2[] = "web.server2.net";
EthernetClient client1;
EthernetClient client2;
if (client1.connect(serverName1, 80)) {
if (client2.connect(serverName2, 80)) {