Adjust an oven rack to the middle position and heat the oven to 350 degrees. Pulse the bread in a food processor to coarse crumbs, about 10 pulses. (Do not wash the food processor bowl.) Spread the crumbs on a rimmed baking sheet and bake, stirring occasionally, until golden brown and dry, 10 to 12 minutes. Let the crumbs cool to room temperature.

Meanwhile, toss the cucumber with ½ teaspoon salt in a colander and let drain for 20 minutes. Combine the drained cucumber, ¾ cup of the yogurt, 2 tablespoons of the scallions, and 1 tablespoon of the cilantro in a bowl and season with salt and black pepper to taste.

Pulse the chickpeas in the food processor to a coarse puree with large pieces remaining, about 8 pulses. In a medium bowl, whisk the eggs, 2 tablespoons of the oil, garam masala, cayenne, and ⅛ teaspoon salt together. Stir in the toasted bread crumbs, remaining ½ cup yogurt, remaining scallions, remaining 3 tablespoons cilantro, processed chickpeas, and shallot until just combined. Divide the chickpea mixture into 6 equal portions, about ½ cup each, and lightly pack into 1-inch-thick cakes.

Heat 2 teaspoons more oil in a 12-inch nonstick skillet over medium heat until shimmering. Carefully lay half of the cakes in the skillet and cook until well browned on both sides, 8 to 10 minutes, flipping them halfway through.

Transfer the cakes to a plate and tent loosely with foil. Return the skillet to medium heat and repeat with the remaining 2 teaspoons oil and remaining cakes. Serve with the cucumber-yogurt sauce and lime wedges.