Put the pepper in a heavy-bottomed pan over high heat. Turn the pepper as the skin begins to blacken and blister on each side. When blistered on all sides, remove from the pan and put in a plastic bag with a few drops of water. Seal the bag immediately with plenty of air trapped inside. Wait 5 minutes. Remove from the bag, cut out the seeds, remove the skin, and dice.
Meanwhile, beat the eggs in a small bowl, and add freshly ground black pepper.
Heat a medium nonstick skillet over medium heat.
Add 1 teaspoon coconut oil when hot.
Add half of the egg mixture to the hot pan. As the egg starts to set, add half of the remaining ingredients to one half of the pan.
When fully set, fold half of the egg over the filling, and cook a minute more.
Repeat with the second omelet.