Preheat the oven to 400 degrees F. Heat 2 tablespoons olive oil in a large pot or Dutch oven over medium-high heat. Add the ginger and 2 minced garlic cloves; cook, stirring, until golden, about 1 minute. Add 1 tablespoon curry powder and the tomato paste and cook, stirring, until the ginger and garlic are well coated, about 1 minute.
Add the lentils, carrots, leeks and 7 cups water. Bring to a boil, then reduce the heat so the liquid simmers. Season with 3/4 teaspoon salt and a few grinds of pepper. Cook until the lentils are tender and start breaking apart, about 25 minutes. Add the spinach and stir until wilted; season with salt and pepper.
Meanwhile, cut the naan into 1-inch pieces. Whisk the remaining 2 tablespoons olive oil, 1 minced garlic clove and 1 teaspoon curry powder and 1/4 teaspoon salt in a large bowl. Add the naan and toss to coat. Spread on a baking sheet and bake until golden and crisp, about 10 minutes.
Top each serving of soup with the naan croutons, a dollop of yogurt and some cilantro.