Place cucumber in a strainer, sprinkle with salt and let sit for 20 minutes. Press out liquid.
Meanwhile, in a large resealable plastic bag, combine the EVOO, half of the lemon juice, three-quarters of the garlic, the parsley and oregano. Season the chicken with salt and pepper, drop into the bag and let marinate for 10 minutes while you preheat 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, the yogurt, the remaining lemon juice and garlic, the mint, coriander and cumin; season with pepper.
Splash a little water on the griddle and add the naan 2 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.