Heat the oven to 400°F. Toss dry chickpeas with olive oil and 1/4 teaspoon salt on a rimmed sheet and roast for 20 to 25 minutes until dry outside and lightly golden, keeping the centers tender rather than roasting them hard.
Cool chickpeas for 20 minutes, then pulse briefly in a food processor until coarsely broken into small pieces, not a puree. Transfer to a bowl.
Stir yogurt, lemon juice, Dijon, dill, remaining salt and pepper together. Fold into chickpeas with dry cucumber and diced radishes.
Cover and refrigerate for 30 minutes until cold; prep time includes cooling and chilling.
Lay dry lettuce over 4 bread slices and divide chickpea salad over it, pressing lightly into even layers. Close with remaining bread and cut in half.
Serve cold or refrigerate at 40°F or below, transporting with ice packs.