I have been trying to achieve "one goggle in the water" side breathing, but even using flippers and three kick boards to keep my body as high as humanly possible, basic face anatomy dictates that if my eye is underwater, half of my mouth is underwater. Now I read that "Your arms create pockets of air near your armpits as you reach ahead and stroke. Swimming coaches teach students to turn their heads to one side and draw a breath from the pocket." Does this mean that the kick board approach is impossible and the only way to practice side breathing is while you are doing strokes? I had thought it was the wake from your head that created the air pocket, which is why I was using flippers to move forward as fast as possible while holding the kick board.
What the coaches mean, is that when you turn your head to the side, water forms kind of a "bow wave" coming off of your head/forehead, which causes a small trough (area of lower water) to form near your nose/chin, and it's in this pocket that you breathe. Yes, your eye will be underwater, but the bow wave moves the water away from the rest of your face.
Speed does help, but once you get used to it, you can have your mouth half full of water and still be able to breathe. As you get faster and faster, the bow wave effect is easier to do and clears your nose/mouth much better.
Use one kick board, and when you want to breathe, drop your hand off the board on that side, pull, then breathe during the arm recovery. This will help to develop your timing and encourage the body roll. You don't want to just turn your head only.
There are various breathing exercises for beginners with the kickboard: If you are right handed and prefer breathing from the right, then keep your left hand outstretched to the front of the board and the right hand bent and on the bottom side, then breathe from the right with normal kicking patterns by twisting your head to the right. The key here is to try to not tilt the heard upwards, only twist it to the right, at that time, half of your face will be outside the water and you can then slowly breathe. If you're left handed, then just switch sides.