hm. So in order not to guess, I made the accurate graph a such a probe. All my apologies but at 20°c it was not 6000ohms but 60200. The typical resistance at 25c is 48 000 ohms. I will find a 48 000 ohms resistance in order to check the display on the PID.

I continued my investigation in order to understand and I think I'm progressing.
As I first bought a 10k NTC @ 25°c I plugged it again. It showed 51 to 55°C onn th PID which is rather logical due to its too high resistance (red curve)
I made a curve of the 3,3K @ 25°c thermistor and it shows that when connected at 25°C the PID reads 140°C or so (grey curve)
The blue curve is fot the NTC 3,3K but at 100°C. it confirms what the seller of the PID measured at about 20°C taht is to say 63k.
My conclusion is that the indication on the PID enclosure must be red not for 25°c but for 100°c. So the typical resistance at 25°c which I thought was a rule is at 48 000 Ohms.
On the diagram, the red dot represent the wrong display on the PID :
- more than 130 with a 3k3 at 25°c
- 51°c with a 10k NTC at 25°c

The green dots are the two measurement for a 3,3k NTC at 100°c = a 48k NTC at 25°c.

You're doing it in a hit or miss fashion. Follow Jeff's advice. It's the easiest, least expensive way.

Not really in a hit and miss. I don't say I'am absolutly right about the statement but the curves are right. these curves are not drawn "by hand". They are caclulated with scientific NTC software at the lab. I followed Jeff advice and each value is simulated at 23 different temperatures with a 5 degree step. R1​=R0​eβ(T1−1​−T0−1​)
It's interesting to see that they are really not parallel.

It's the way you're figuring it out that seems a little wonky. The resistance of a thermistor changes with temperature so you won't know exactly what it is when it's plugged into the PID. It would be so much simpler to use a few previously measured resistors. Way easier. Well, that's what I'd do anyway.

Knowing why an information given by Gicar (3K3) does not match with the usual given 25°c does not sound really wonky to me. I like to understand things.

I didn't mean any offense. Just that if it were me I'd have already plotted the R vs T curve and ordered the thermistor. Last week. But it's a DIY project not a DITWJPWDI project.

No offense, don't worry. It's just that I'm on a long time project and not in a hurry so I have te time to investigate. Don't worry I have another working espresso machine

So, here are some results.
I first tried an 47k thermistor that I had aside and the result were bad...
I ordered a 33k one and it's much better. I have a 2 degree delta compared to my kitchen thermometer. That said, not sure that a 10 euros Ikea thermometer is that accurate.
So know I'm pretty sure that the probe mentionned before is the good one. This PID needs a 33k thermistor. I found a 33k resistor at home and the value at 25°c is good.
Nevertheless, the delta may be explained by the Beta difference. On the one I chose, the beta is 4090 and I think it should be 3970.
Here are the different curves with the two betas. Still, I don't understand the 3k3 reading on the enclosure... It make no sense to me.