Unfortunately there's no easy and accurate way to measure the temperature of the water exiting the showerscreen.

The reasons for this are many and varied but essentially anything that the water droplets/stream hit before you measure the temperature will cool it - including the air it is falling through, any cup / receptacle that it is falling into and any thermometer. you are using to measure

Trying to use an infrared thermometer on a stream of water will also give a largely inaccurate temperature reading - and similarly trying to use one with water in a cup or other receptacle will also not work well.  The thermal emissivity of water / droplets / steams for using an infrared thermometer makes this very difficult if not impossible.

Using any of the above methods you would often see a temperature that is 10c or more less than the actual water temperature exiting the showerhead.   So measuring brew water temperature using any "normal" method would often result in a reading of high 70s or low 80s celcius - with brew water that is actually in excess of 90c.

Many other standard (E61 style) machines can utilise an expensive portafilter type pressure and temperature measuring device (called a Scace 2) to determine the brew pressure and temperature more accurately - and although these aren't available for Gaggia group heads and E61 style one can fit to an extent and be used - though the cost is prohibitive for the normal home user (£500+ if you can find one).

Should you wish to try and measure the temperature of water exiting the Gaggia Classic showerscreen then the "flash to steam" method gives reasonable results.  This relies on the fact that water at 100c will be steam, and water at 99c will still be liquid.

With a PID you will have an "offset" (Psb in menu 0089) configured.  This is typically around -8.0c.

If you set your PID to a displayed set temp (SV) of 97c and leave it to stabilise for around 5 minutes then you should find that when you press the brew switch, the water that exits the shower screen initially is liquid water.

If you then adjust the PID by 1c, to 98c, and again leave for 5 minutes to stabilise you should find, again, that when you press the brew switch the water that exits the shower screen initially is liquid water.

Continuing the above process (of 1c increments and waiting 5mins between tests) you SHOULD find that at around 99/100/101c when you press the brew switch the initial water that exits the shower screen goes to steam.  

You can then use this temperature to more specifically tune your offset to your machine.

If you find water flashing to steam only at 102c then your offset needs to be changed from -8.0c to -10.0c.   If you find water flashing to steam at 98c then your offset needs to be changed from -8.0c to -6.0c. etc.

Re-establishing your offset using the above 'flash to steam' method will enable you to have your 'flash to steam' temperature point recalibrated to 100c on your PID for your machine.

Resetting your offset (Psb) using a vague or incorrect thermometer temperature reading obtained from water in a cup, or falling through the air, will result in you massively changing your Psb to an incorrect value that will result in your brew water probably being much much too hot.

Hopefully this helps - but the basic message is, unless you REALLY know what you are doing and have the right tools or skills to do so - measuring brew water temperature accurately is very very difficult indeed.