Update: It appears that this requires 5 pieces of the Torug craft set. It's not just 5 pieces of heavy armor as I thought, but specifically 5 pieces of Torug. ("reduce weapon enchantment internal cooldown" This set bonus is getting fixed with Craglorn. Although the thread will soon be defunct I do try to recover the dps loss further in the thread if you are interested...
There are at least 4 ways to increase your attack speed while using a Two Hander.

1)Base attack speed.
Craft your 2 hander with the "weighted" trait. (A purple weapon gives you +10% attack speed)

The other options take timing and finesse. They are combo moves.

2)Heavy + Instant attack combo.
This move gives you a full heavy attack and a skill attack all in one second with only a 1/4 of the heavy attack animation. Put executioner or another instant skill on your bar. Charge your heavy attack till the sword is wound up fully then release it. After you release it immediately pop your instant skill. You will release a full heavy attack and your skill attack almost at the same instant while avoiding the tail end of the heavy attack animation.

3)Medium heavy + Light attack combo.
Normally a light attack will have to pass from the far right, to the target for damage, and then there is a relief animation to the left while the swing momentum is stopped. Or in other words a light attack is a 180 degree swing. We are going to make that a 90 degree swing and add a medium heavy attack to it.

For this attack Charge up your heavy attack, but instead of waiting till it's fully wound up release it early and immediately pop the left mouse button. You will do a cross attack, combing a medium heavy and a light attack almost at the same instant and avoid the 90 degree momentum release animation.

4) Medium heavy + Light attack Combo Chain! First master combo 3 so that you can perform it at will. Once you get that down you can chain that same combo together and avoid the animation.

1) Charge your heavy just a bit
2) release and then pop your light attack.
(so now your weapon has moved from the right to the left because of the medium heavy attack and it's now following back to the right due to the second light attack.
3) Just after your second attack passes center as it swings from left to right start charging a heavy attack and repeat the process. (if you didn't add this chain the second attack would go through it's cooldown animation where you are releasing the momentum behind you.

This is a very powerful combo chain. Use it till the mob is half health and then switch to combo 2.

3 Hit Combo - (Full Heavy + Medium Heavy + Light)
1) Fully Charge your heavy attack, Hold until the sword winds all the way back and begins to roll over the top.
2) Release the left mouse button and hit it twice more.
3) You will do a very fast 3 strike hit. Looks pretty awesome too.

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Once you get the hang of this you will nearly double your attack speed. If you are having trouble with combo 3 turn your sound up so you can hear the hits. You want a quick tap.tap..tap.tap, not a tap.....tap..... Think of a normal attack swing as a 180 degree swing. 90degrees of that swing is a waste as it's already hit the mob. When you hit the light attack after the medium attack you are reversing the direction of your post hit swing. As you get better the degree between 180 and 90 will get smaller and smaller. You can even get it to around 45.

Another way to envision it is imagine you are swinging a sword. You charge up, you hit the opponent, and as soon as your blade flies past your opponent you twist your wrist and reverse direction. This is what you are doing in the combo by using the left/right mouse buttons and the right timing.

I've also got a short video below showing the difference between using these combos and just spamming attacks. ESO is so much more complex than just left click mouse spamming. The first video is the main video, shot on high resolution so that you can see the combat log. However due to the resolution it's choppy and the audio doesn't quite sync up. I made a second video to help display the animation.


The second video is to show case the animation for the 2 hit combo chain, shot at a lower rez so you can see it better.


The third video is to show case the animation for the 3 hit combo, I forgot to lower the rez so audio is out of sync and the attacks are a bit choppy. You can still tell by the combat log and speed whats happening. This is a good combo for opening from stealth, or for when an enemy is stunned.