Place the cucumber in a strainer. Sprinkle with salt and let sit for 20 minutes. Press out the liquid.
Meanwhile, in a large resealable plastic bag, combine the EVOO, half of the lemon juice, three-quarters of the garlic, the parsley and the oregano. Season the chicken with salt and pepper; drop into the bag and let marinate for 10 minutes while you pre-heat the broiler.
Scatter the chicken across a broiler pan and broil, turning once, until charred and cooked through, 10-12 minutes.
Heat a griddle over medium-high heat. In a food processor, puree the drained cucumber, yogurt, remaining lemon juice, garlic, mint, coriander and cumin; season with pepper.
Splash a little water on the griddle and add the naan, two at a time. Cook, turning once, to crisp up, about 2-4 minutes.
Pile the chicken onto the naan and top with the fixins and tzatziki sauce.