You should always shoot a shotgun with both eyes open. Why? Well, for two reasons, depth perception and field of view.
But wait, you say, "I am cross dominant and cannot shoot with both eyes open".
Yes, you can. But you have to mount and sight in a reflex, red dot or Holosight. Once you have one of these parallex free sights on your shotgun, not only can you shoot with both eyes open, but you can shoot from either shoulder with both eyes open.
I speak from experience. I started with the Holosight but changed to the Trijicon Reflex II because of fear that the Holosight batteries would go out on me at an inoppertune moment. Trijicon now produces a Tri-power reflex sight that may be best for those who shoot from within a fully enclosed blind because of the option to use a battery setting in addition to the tritium and ambient light options.
Your wingshooting (doves & ducks) will also improve as you become comfortable with the sight as well.
Cross-dominance. How can you tell if you suffer from this malady? I believe that you can take a paper towel cardboard insert, pick a target at about 15-20 yards and without thinking, raise the cardboard tube to your "eye" to look at the object and you will, in most cases, bring the tube to your dominant eye.
Another method is to place your hands together at arms length and make a hole with your thumbs and forefingers and look at the target. Bring your hands to your "eye" and that will be your dominant eye.
As stated above, sometimes your dominant eye will want to change or waver. The use of the reflex, red dot or Holosite will make that situation irrelevant to your shooting with both eyes. God gave you two eyes for a reason. Use them.
Oh, and JPH, it is broke. I am cross dominant, right handed with a left master eye. Trust me on this one and you will thank me. I find it hard to miss a dove anymore and I have not missed a turkey in (Knock on wood) ten years.
Why be good when you can B-Mobile or Spin & Strut.