Pierce each potato with a fork and cook in microwave for 5 minutes per each potato. Scoop out potato from skin and transfer to a food processor or high quality blender.
Combine drained and rinsed chickpeas, lemon juice, tahini, oil, cumin and garlic in the food processor. Puree, about 1 minute; thin with water if necessary. Season with salt and pepper and let cool.
Cut broccoli into smaller florets, peel and cut carrots into sticks and cut pepper into strips.