Serving binary file with ESP8266WebServer
It looks like you can use the send_P function to send raw PROGMEM data:

void send_P(int code, PGM_P content_type, PGM_P content, size_t contentLength);

I.e., (from what I can gather):

PGM_P favicon = {
  0x00, 0x00, 0x01, 0x00, 0x01, 0x00, 0x10, 0x10, 0x00, 0x00, 0x01, 0x00,
  0x18, 0x00, 0x68, 0x03, 0x00,......... 0x00
};

server.send_P(200, "image/x-icon", favicon, sizeof(favicon));


Incidentally, PGM_P is just a typedef for const char *.