Preheat the oven to 375°F. Stack the pitas and cut each into 8 wedges. Separate the wedges and arrange in even layers on two large rimmed baking sheets. Spray the wedges with nonstick cooking spray. Bake for 12 to 15 minutes, until golden and crisp. Remove from the oven and let cool completely.
Preheat the broiler. Place the poblano chile on a small rimmed baking sheet. Broil for 10 to 15 minutes, turning the chile occasionally, until charred on all sides. Transfer the chile to a small resealable bag and let rest for 20 minutes. When cool enough the handle, peel the chiles, remove the seeds, and roughly chop.
Combine the poblano chile, chickpeas, garlic, salt, lemon juice, and tahini in the bowl of a food processor and process until smooth. Transfer the hummus to a serving bowl and drizzle the top with olive oil. Serve with the pita chips for dipping.