In a small skillet over medium heat, add 1 teaspoon butter and halved jalapeno. Cook until jalapeno is soft and charred in spots, maybe 3-4 minutes. Remove jalapeno, remove seeds from pepper, and chop.
In a small bowl, mash together cheddar cheese, chopped pimento peppers, and jalapeno. In a separate bowl whisk together 2 eggs per omelet.
In a small (8-inch works) omelet skillet, add 1 teaspoon butter over medium heat. Once melted add 2 whisked eggs. As the eggs cook, use a spatula to push the cooked egg to the center. Tilt the pan a bit to let the uncooked eggs flow outside.
When most of the egg liquid is cooked, there still might be a thin uncooked layer on top, add half of the filling mixture to half of the omelet. Fold the omelet over and cook for 15-30 seconds on each side until the eggs are set and the cheese is starting to melt out of the omelet.
Slide omelet out onto a plate and garnish with scallions. Season with salt and pepper and eat while warm.