Mix rosemary, black pepper, salt, oregano, garlic powder, onion powder, cardamom, and coriander together in a small bowl. Place chicken on a plate and season with spice mixture. Heat vegetable oil in a saucepan over medium heat until it starts to shimmer, 2 to 3 minutes. Add seasoned chicken to the pan and cook, covered, 4 to 5 minutes. Flip chicken over and cook, uncovered, until browned on the outside and no longer pink on the inside, 5 to 6 minutes more. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Squeeze lemon juice over chicken and allow to cool 5 minutes. Slice into strips.

While chicken cooks, bring water to a boil in a saucepan and add salt to taste. Add couscous, stir once, and cover. Remove saucepan from heat and let couscous steam for 5 minutes. Fluff with a fork.

Mix cucumber, sour cream, yogurt, lemon juice, olive oil, garlic, mint, dill, salt, and black pepper together in a bowl until tzatziki sauce is just combined.

Divide cooked couscous between 4 bowls. Top with cooked chicken, tzatziki sauce, broccoli, onion, cucumber, tomatoes, olives, parsley, and feta cheese. Serve immediately or store in airtight containers to reheat later.