Combine the lamb or turkey with the spinach, yogurt, garam masala, garlic and finely chopped mint. Season with salt and pepper. Place about one-sixth of the mixture on a piece of wax paper or parchment and place another sheet on top. Roll out or press the mixture into a thin, free-form patty no more than 1/2 inch thick. Repeat with the remaining mixture, making 6 patties total.
Combine the coarsely chopped mint with the tomatoes and red onion. Season to taste with salt and pepper.
Heat a griddle to medium-high. Drizzle with EVOO to lightly coat and cook the patties until golden and cooked through, 3 or 4 minutes on each side. Wipe the griddle, add a splash of water, then cook the naans, a few at a time, until crispy and blistered on each side. Halve the naans. Divide the patties among 6 naan halves. Top each with some tomato salad, a dollop of yogurt and another naan half.