How to solve EOCross efficiently
By yoruba
For the mainstream methods, a strategy for solving intuitive steps is to break them into 2 easier to manage steps. Then as you get more skilled, you will begin to find ways to merge both of these steps into one.
In the CFOP method for example, less experienced solvers might plan 23 cross edges at once, and then solve the rest of the cross. Or in the Roux method with the first block, solvers choose between various approaches depending on the scramble: mainly square+pair, line+other line or F2L style.
In ZZ, the main approaches are:

EO+2 > the rest. This is where you solve your EO and any of the 2 cross edges at the same time, and then solve the rest of the cross edges. The EO+2 step should influence the last 2 edges as much as possible (most commonly used, for 0, 2, 4, and 6flips).
Example 1 (opens in a new tab), Example 2 (opens in a new tab) 
Partial EO+2 > solve the other 2 while finishing EO. In this approach, you orient some, but not all of the edges while solving 2 of the cross edges, and then you orient the remaining edges and solve the remaining cross pieces. (Most commonly used for higher nflips: with 6+ bad edges).
Example (opens in a new tab), Example 2 (opens in a new tab) 
Cross > EO. Here you solve the cross pieces first, and then orient the edges. (This is by far used the rarest, and is only really used in 2flips from time to time when you can cancel into a sledge at the end).
Example (opens in a new tab)
Influencing
Influencing is about affecting the cross pieces during the EO solution, to get a better cross. You should learn these fundamental influencing techniques:

Using different directions for F/B moves

Easiest example: Solving the edge directly:
Example (opens in a new tab) 
Inserting other cross edges:
Example 1 (opens in a new tab), Example 2 (opens in a new tab)


Switching the bad edges you're going to orient
Example (opens in a new tab) 
Doing extra moves before an EO turn (replacement technique)
Example 1 (opens in a new tab), Example 2 (opens in a new tab) 
Handling cases with 2 bad edges on the F/B faces, where you manipulate the position of EO. For this Learn how EO gets changed, as well how it affects the cross pieces

Adjacent 2 bad edges:
Example 1 (opens in a new tab), Example 2 (opens in a new tab) 
Opposite 2 bad edges:
Example 1 (opens in a new tab), Example 2 (opens in a new tab)


Solving EO differently to influence cross pieces:
Example 1 (opens in a new tab), Example 2 (opens in a new tab)
Movecount
We've gotten through the theory and things you can do, it's time to implement them in practice. For this, we will use the Movecount Drill. It has helped many people and the writer of this article used it singlehandedly drop his movecount from 10.5 to 8.5.
First, set up your CSTimer settings:
 Go to CSTimer (opens in a new tab).
 Click on the gear in the top left corner.
 Go to the 'Timer' section.
 In the 'entering in times with' box, change the setting from 'timer' to 'typing'.
Now you're ready. The drill itself is very simple:
 Scramble the cube.
 Solve EOCross, and count how many moves it took you. Type that as a time with 2 zeros (so if it took you 15 moves, type 1500).
 Solve the rest of the cube.
If you haven't done this drill, repeat it 50 times to get a sense of your average movecount.
Now that you have your average movecount, you want to set your target movecount. It should be set 2 moves above your average movecount. So if your EOCross averaged 10 moves in a session, your target movecount will be set to 12 moves.
After a certain amount of solves in the session (preferably 12, 25 or 50) you want to go over every scramble above the target movecount. You want to repeat the scramble for these and find a solution that is below that target movecount. This will force you to try different approaches to the hardest scrambles and to apply techniques you have learned. For example, you may try to solve a different set of 2 cross pieces while solving EO, and check if they influence the remaining 2 cross edges to a better finish. Then go over the solution once again to find out why. This sort of curiosity and the growth mindset will allow you to make the best and quickest progress.
As you get better, you can also start checking the EOCrosses that were only 1 move longer than the average, and if you get very good (89 moves on average), you'll know instinctively to go for the scrambles for which the solutions 'felt' bad, even if they had normal length solutions. But if you're a beginner, don't think about it for now.
If you're really stuck, go to Solvers > EOCross to see what the computer will give you (for x2y, check the first 4 solutions). What 2 first edges does it solve, how does it deal with EO, what approach does it use? You can also use these to find other techniques that you haven't considered.
Eventually, after doing the drill a couple of times, your average movecount will drop. Apply a new target movecount for progression, and repeat the drill.
Continue doing this process until your average movecount drops to 8.5. Then, stop caring about movecount and start practicing for speed.
How to solve EOCross as one step
We want to eventually be able to see EOCross as one step to solve, instead of 2 intermediate ones.
If you recall the start of the article, it talked about being able to merge both steps as you get more skilled. The solution is really just more deliberate practice: after getting comfortable with each approach, you will start to see which ones will have the best balance of how short they are, and how much they influence the rest of the pieces and EO. These will be the best to use for a scramble.
If you want more help on this, you can go to crystalcube (opens in a new tab) and train the scrambles with the highest nflips and the longest solutions, and then study the solutions which the computer gives.