To make the spicy roasted chickpeas, preheat oven to 400°F. Spread rinsed chickpeas on a clean kitchen towel and rub to dry. Toss chickpeas with olive oil, cayenne, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Spread in a single layer on rimmed baking sheet and bake for 30 minutes, stirring halfway through baking time. Cool 15 minutes.
In a food processor, combine roasted chickpeas, mayonnaise, tomatoes, peppers, basil, and 1/4 teaspoon salt. Process until smooth, scraping down the sides as necessary.
Meanwhile, whisk together mayonnaise and pesto in a small bowl and set aside.
To assemble the sandwiches, lay 4 flatbreads on a flat surface. Divide the Spicy Chickpea Salad among 2 flatbreads, spreading each with a thick layer. Spread the remaining 2 flatbreads with Pesto Mayonnaise.
Arrange the cucumbers in a single layer on each Chickpea Salad flatbread, then sprinkle with lettuce. Top each with a Pesto Mayonnaise flatbread.
Cut each of the two flatbread in half or into triangles and serve.