Combine the chickpeas, water, cheese, lemon juice, salt and red pepper flakes in a food processor and puree until completely smooth, about 5 minutes. While the motor is running, pour in the olive oil and process until fully incorporated and velvety. Season with pepper, to taste.
Spoon a scant tablespoon bean spread into toasted pitas. (Put remaining bean spread in an air-tight container.) Add avocado and cucumber slices, or other vegetables of choice and season, to taste, with salt and pepper. Wrap in parchment or butcher paper, and pack in a plastic container to keep from getting squished. Pack in a lunch box and send to school.
TIP: Remaining bean spread becomes a dip for vegetables or pita chips, or as another dimension layered in your child's favorite cold cut sandwich.