Make the Dressing: Whisk together all of the ingredients until smooth.
Make the Nachos: Preheat the oven to 400°F. Cut each pita into 8 wedges. Arrange the pita wedges in a single layer on an aluminum foil-lined rimmed baking sheet. Brush the pita wedges with the oil and sprinkle with the paprika and salt. Bake until crisp, 10 to 12 minutes.
Remove the pita wedges from the oven. Increase the heat to a high broil. Sprinkle the wedges with the feta, bell pepper, and red onion. Broil until the cheese starts to melt, about 3 minutes. Remove from the oven.
Sprinkle with the tomato, olives, and chives. Transfer the dressing to a squeeze bottle, or spoon into a plastic ziptop bag and snip 1 corner to make a small hole. Drizzle the nachos with your desired amount of dressing. Reserve any remaining dressing for another use.
To make this more kid-friendly, top the toasted pita with store-bought hummus, feta, and diced avocado.