In a medium bowl, whisk together eggs and egg whites; season with salt and pepper. Whisk in cheese.
In a small nonstick skillet, heat oil over medium-low. Cook eggs, scraping bottom frequently with a flexible heatproof spatula, until just set, 1 to 2 minutes; stir in dill.
Spoon eggs into pita halves, and garnish with more dill.
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