This soup is even better the day after it's made. Thin it with extra water if it thickens overnight. Claudia Roden's version has you caramelize an onion in a few tablespoons of oil while the soup cooks, and if you feel like using an extra pan, I highly recommend this addition. Scoop some onions on top of the soup when you garnish with the croutons.
Heat the oil and onion in a heavy stockpot over medium heat until onion is soft, about 10 minutes. Add the garlic, cumin, coriander, and red pepper flakes and stir. Add the lentils and the carrot, then stir. Add 2 quarts (8 cups) of water. This will look like a ton! Bring to a boil, then lower to a simmer and cook uncovered until the lentils have disintegrated and the soup has a creamy texture, about 45 minutes. Add 1½ teaspoons salt, fresh pepper, and the juice of half the lemon. Taste, then add more salt if needed.
Meanwhile, turn on your broiler. Make the cheesy croutons: Split the pitas in half. Brush a baking sheet with olive oil. Drizzle each pita half with olive oil and sprinkle with za'atar. Broil until toasty, about 2 minutes. Remove from the oven, then distribute the grated cheese among the pitas and bake for 3 more minutes, watching carefully, until the cheese is melted and the pitas are crisp. Remove from the oven and cut into uneven wedges.
Serve the soup very hot, with lemon wedges and croutons.