Preheat oven to 350 degrees F (175 degrees C).

Heat the butter in a skillet over medium heat until the foam subsides, and cook and stir the onion and tomato until the onion is translucent and the liquid from the tomato has evaporated, about 8 minutes. Transfer the tomato and onion to a bowl. Pour the beaten eggs into the cooked vegetables, and stir in tarragon, basil, salt, black pepper, and Cheddar cheese until thoroughly combined. Pour the egg mixture into the red bell pepper, and set into a small baking dish, standing upright. Cover the dish and pepper with foil.

Bake in the preheated oven until the pepper is tender and the eggs are set, about 55 minutes.