Spread cream cheese on toast; sprinkle with chives. Set aside.
Heat 1/4 teaspoon oil in a small skillet over medium heat. Add ham and cook until warm, about 15 seconds on each side. Transfer to a plate and cover to keep warm.
Whisk egg in a small bowl with pepper. Heat the remaining 1/4 teaspoon oil in the small skillet over medium heat. Add egg; cook, stirring, until soft curds form and are just moist, about 30 seconds.
Top the toast, cream cheese-side up, with the ham, tomato and the egg. Whisk lemon juice and mustard in a small bowl. Add sprouts and toss to coat. Top the sandwich with the sprout mixture.
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