Preheat the broiler. Soak the chopped onion in a bowl of cold water while you prepare the shrimp.
Toss the shrimp with the olive oil, paprika, 1/4 teaspoon salt, and pepper to taste on a rimmed baking sheet. Spread in an even layer and broil until the shrimp are opaque, about 4 minutes. Transfer to a bowl and let cool. Wash and dry the baking sheet.
Drain the onion and add to the shrimp along with the yogurt, carrot, cucumber, dill, and salt and pepper to taste. Grate in 1 teaspoon lemon zest and squeeze in about half of the lemon juice.
Lay the pitas on the baking sheet and toast under the broiler. Cut the pitas in half and fill with the lettuce and shrimp salad.