Place chicken in a pot and cover with water. Add salt, garlic powder, onion powder, and juice of 1 lime.
Bring to a boil, then reduce to a gentle simmer. Cook until the chicken reaches 165°F in the thickest part, about 15–18 minutes.
Remove chicken and cool for 10–15 minutes. Chop into small ¼- to ½-inch pieces.
Finely chop onion, bell pepper, cucumber, and tomato. Remove cucumber seeds and most tomato gel. Pat vegetables dry if needed.
Combine chicken and vegetables in a large bowl.
Add ¾ cup mayonnaise, lime juice, parsley, salt, and pepper. Stir gently.
Taste and adjust with more mayonnaise, lime, salt, or pepper as needed.
Cover and refrigerate for at least 30 minutes.
Place dry lettuce on bread slices. Add about ½–¾ cup chicken salad and spread evenly.
Top with another bread slice, cut diagonally, and serve.