Preheat the broiler. Line 2 baking sheets with foil. Combine 1/4 cup yogurt with the tandoori paste in a medium bowl. Add the chicken, season with salt and pepper and toss well. Set aside for 10 minutes. Toss the grated cucumber with 1/4 teaspoon salt in a colander; set aside to drain.
Butter the naan on one side and sprinkle with the garam masala. Place on one of the baking sheets and broil until lightly toasted, about 2 minutes. Arrange the chicken on the other baking sheet and broil until cooked through and starting to brown, about 8 minutes. Transfer to a cutting board and let cool, then chop.
While the chicken cooks, squeeze the cucumber with your hands to remove the excess moisture. Mix the cucumber with the herbs, lemon juice, remaining 3/4 cup yogurt and 1 tablespoon water; season with salt and pepper.
Scatter the chicken, cheese and red onion evenly over the naan. Broil until lightly browned and the cheese softens, 1 to 2 minutes. Spoon the cucumber yogurt over the pizzas. Top with more herbs.