That's very unusual. Your brain needs ready supplies of glucose to function. Not eating breakfast is forcing your body to turn energy stores into glucose in a way that it tries to avoid and probably doesn't give your brain ample supply, or just not function well.
Do you actually work better, or have you decided that you work better?
Many well functioning people, including IIRC navy attack divers, live on a glucose free diet for a long time, so I am not alone.
As for if I work better or just have decided I work better, I only have 4 or 5 years of experience with this but despite the fact that I love breakfast and lunch I still skip them because the brain fog I seem to get between 1200 and 1400 on days when I eat breakfast just isn't worth it.
My comment is only pointing out that for me it is the opposite: I work better without breakfast.