Heat 3 to 4 tablespoons olive oil in a large skillet over medium-high heat.
Add the chicken pieces and evenly sprinkle with 1½ tsp cumin, 1½ tsp smoked paprika, 1 tsp turmeric, 1 tsp salt, and 1 tsp black pepper; cook for about 5 minutes, turning occasionally so all sides brown.
When the chicken is about 90% done, reduce heat to medium-low, add 3–4 minced garlic cloves, and cook 1–2 minutes until fragrant, stirring occasionally.
Carefully pour in 1/4 cup lemon juice (it may bubble) and simmer 1–2 more minutes until chicken is cooked through. Remove chicken with a slotted spoon to a bowl, reserving pan juices.
Add the drained chickpeas to the skillet with the reserved juices; season with salt and pepper if desired and add 1–2 tablespoons olive oil only if needed. Cook over medium heat about 5 minutes, stirring occasionally, then transfer chickpeas to the bowl with the chicken, including any pan juices.
To the bowl with chicken and chickpeas, add 2 cups tomatoes, 2 cups diced English cucumber, optional 1/2 cup diced red onion, and 2–3 tablespoons minced parsley; toss gently to combine.
Make the sauce: whisk together 6 ounces plain Greek yogurt, 2–3 tablespoons lemon juice, 1 minced garlic clove (optional), 1 tsp cumin, 1/2 tsp ground coriander, 3/4 tsp salt, 3/4 tsp black pepper, and 1/2 tsp sugar (optional) until smooth. Taste and adjust seasoning.
Drizzle the yogurt sauce evenly over the salad bowls to taste and serve immediately.