Heat oil in a medium skillet over medium heat. Cook broccoli and scallion covered, stirring occasionally, until tender, about 4 minutes. Season with 1/8 teaspoon salt and some pepper. Transfer to a plate.
Add cheese to skillet. Cook over medium-high heat until melted and golden, about 1 minute. Reduce heat to medium-low. Add egg and egg whites. Keeping egg yolk intact, drag a spatula through the whites to spread them evenly in the pan. Cook, covered, until whites are set and yolk is still runny, 1 to 1 1/2 minutes. Top with broccoli-scallion mixture, and fold omelet over. Sprinkle with a pinch of red-pepper flakes.
Serving Suggestion: To work whole grains into the breakfast, serve with one or two slices of whole-wheat toast.