Headphone Impedance explained
This requires good understanding of a thing called Damping Factor.
Mobile phone ear buds, and large, standing loudspeakers work by the same physics.
Imagine, you have a loud piece of heavy music playing, and you can see the bass membrane moving wildly. Then suddenly the music stops. Now, the bass membrane at that very moment was heavily shaking, but since no more signal is applied, soon after it will stop moving. The question is now how soon after. Most of all, the membrane was still moving for a tiny fraction of a second, while there was no signal applied. During this short period of time, the membrane did NOT represent the music signal. It was making movements all by itself, which the amplifier did not say it should do so. This is distortion, which is defined as a signal at the output (the membrane), which was not applied at the input (the speaker terminals).
In a practical situation, it is not so bad, because the speaker is also damped by the amplifier. That is because the speaker, if moving without signal applied, becomes a voltage generator. If the membrane is moved, at the speaker terminals would appear exactly the same signal, as normally would be needed to make this movement possible. So what happens, if the music stops all of a sudden? The still moving speaker responds with a generator signal which signal is now shorted by the amplifier output. Remember the amplifier output has zero volts on it. Zero Volts is the same a short. This short absorbs the energy, and the membrane will stop moving after all energy is gone.
The damping factor.
The amplifier output damps those unwanted movements. The speaker outputs will have a specific impedance. This impedance is much lower as the speaker impedance! So when your tube amplifier has "8 Ohms" written at the speaker terminals, this only means you should connect 8 Ohms speakers to it. It does not mean the output impedance is 8 Ohms. It is much lower, and that is for good reason. The speakers itself are actually 8 Ohms indeed. In case of zero signal suddenly, the movement energy of the speakers is much faster destroyed when the speaker output is 2Ohms, as when it is 8 Ohms. It's exactly this, what amplifier manufacturers are doing.
This faster way to stop the movement, is captured with a number, called: Damping Factor. It is the ratio of speaker impedance and output impedance. In this example 8 divided by 2. So the damping factor is four. After having understood, how a high damping factor reduces unwanted speaker movements, we have to realize a new movement of the speaker follows the same pattern. Initially, there is electrical signal, but the movement will have some delay. Here too, there is a definition of distortion, because the membrane follows the signal not exactly. WIth a damping factor or four, the speaker becomes easier to drive with 2 Ohms, as with 8 Ohms.
The conclusion is, the higher the damping factor, the more precise the membrane movement will equal the electrical signal. Which is what we want of course.
Size plays no role. Just with a smaller speaker we can't see the movements any more. WIth smaller speakers, the whole principle is however exactly the same, and head phones are not excepted from these laws of physics.
What does it come down to, how does a too low damping factor sound? It is perceived by the ear as a muddy, blurry sound. Whereas a high damping factor is described by many as natural sounding, precise and holographic.
So what to do now?
Very simple. Find out the impedance of the head phone, and connect it to an output with a much lower impedance. The ratio of those two numbers is.... The damping factor :)
The output impedance of mobile phone can be for instance 50 Ohms. So a 300 Ohms head phone will probably sound good on it. This is a damping factor 6. Or, when you want to build your own amplifier for a 300 Ohms head phone, select a tube from the tables which can give a one or two impedances above and below 50 Ohms. So it doesn't have to be EXACTLY 50 Ohms. The damping factor is mandatory anyway. If we would have taken 5x or 7x, instead of 6, we would have had a different outcome. Simply try out which is the setting that sounds best, by reprogramming the EE29 PCB. There is no better way as simply trying it out.
A good choice would for instance be the LL2765, and use it 6:1 with the EML 45.
Do not forget to try out 12:1 and 4:1 as well.
A FInal word: At least, hope it is clear now, why it is so essentially wrong to connect a 300 Ohms head phone to a 300 Ohms output.