This looks "burnt" and is also the typical location of the thermal fuse.



How about a pic of the boiler top?

The "burned" spot is just black tape on the wire. Here is a little better photo with the cable pulled away from the boiler, showing there is nothing on the boiler under the "burned" spot.



And here is a photo of the top of the boiler:

That is the thermal fuse.



You may want to check it for continuity, i.e., essentially 0 ohms.

From the description, and just guessing, it sounds like either the switch or control box or bad connection self destructed when powered. If the element reads ~40r, the resistance across the power leads with the power on and element connecter should be slightly less than the element only.

What tube did the water pour out?
That in itself is very strange, unless it's the OPV bypass.
In that case the control box is the likely culprit : It's running the pump, but not opening the boiler fill solenoid.
see Eric's excellent Millenium electrical diagram for control reference

I checked resistance across the thermal fuse and it was zero Ohms. The black tape is very brittle (why put tape on a connection at the side of a boiler??).

The resistance across the heating element was 39.7 Ohms and the resistance across the power leads with the switch on and the heating element connected was much higher--about 2400 Ohms. Not sure what that means.

I am not sure which tube the water came out of, the water suction tube or the recirculating water tube--I shut the unit off as soon as I could after the pump started.

By the way, the label on the control box on my machine says "PRO.EL.IND". Does not seem to be the Glemme controller shown on Eric's diagram. Maybe because my unit is 230V?

why put tape on a connection at the side of a boiler?

Because the thermal fuse connections are protected by only one terminal insulator and the red covering which, IIRC can slide around easily save for the tape - its been a while since I worked on one.

Your pump's system (and lots of other machines) is fitted with a "air release" valve:



This valve will discharge water at a small rate until brew pressure reaches 6-8 bar to allow the pump a greater chance of dispelling any air that might get in the pump's suction line when doing maintenance or initial operation. That is where your water is coming from.

I believe you need to do a non-damaging autopsy of the black control box - remove the sheathing and plastic cover to search for damage. The company that made that box is a Giemme competitor. As Ian sorta hinted, you will likely see some electrical damage inside - likely the heating element relay that is controlled by the pstat.

Your machine looks sorta new - is it still under a warranty?

Now take this with a grain of coffee , that control box has a history of bad acting in other machines. I might have mentioned this earlier had I recognized the box.

Ok, thanks for all your help! I will look into the controller tomorrow. Maybe I should just buy a new controller? Then if that is not the problem I will have a spare.

No - first you need to diagnose the problem and (hopefully) what caused it or else you may witness a repeat performance.