1. Place the chicken breast fillets in a large freezer bag and add 2 tablespoons of the oil, the lemon zest and juice, the garlic, half the fresh oregano and all the dried oregano and toss together well, ensuring the chicken is well coated. Seal the bag and set aside in the fridge to marinate for 1-2 hours, or overnight if possible.
2. Meanwhile, for the dressing, stir together the lemon juice, Greek yogurt, mayonnaise, tahini and garlic. Transfer to a small serving bowl and sprinkle over the sumac. Chill until ready to serve.
3. Heat a griddle or frying pan and cook the chicken for 15-20 minutes, turning occasionally, until golden and cooked through with no pink meat. Transfer to a board and cover to keep warm.
4. Toss the broccoli in the remaining tablespoon of the olive oil and place on the griddle pan. Cook for 4-5 minutes, covered loosely with foil, until tender and charred.
5. Slice the chicken and sprinkle with the remaining fresh oregano, then serve with the broccoli, garlic-tahini sauce and new potatoes.