In a small bowl, stir together the goat cheese, chives, and lemon zest. Using a vegetable peeler, shave asparagus spears into long, thin shavings. Place shavings in a small bowl. Drizzle a little of the olive oil over the asparagus shavings and gently toss.
Drizzle a little olive oil over the flatbread. Place dollops of goat cheese evenly over the flatbread. Top with shaved asparagus. Season with salt and black pepper, to taste.
Place flatbread on a large baking sheet and bake for 12-15 minutes or until cheese is melted and flatbread is crisp. Remove from oven and top with chopped pistachios and lemon zest. Serve immediately.