Slice off tops of peppers about 1/2 inch down from stems, reserving tops for another use if desired. Remove and discard seeds and cores. If any won't stand upright, shave a little off bottom, making sure not to cut into cavity; set aside.
Preheat oven to 350 degrees. Cut leek into quarters lengthwise; slice 1/4 inch thick Place in a bowl of cold water; soak 5 to 10 minutes to remove dirt and sand. Lift leek slices out of water; discard sandy water. Drain in colander; set aside.
Heat oil in a large nonstick skillet over medium heat. Add leek and garlic. Cook, stirring occasionally, until they begin to often, about 5 minutes. Add carrot and poblano pepper; cook until just soft, about 5 minutes. Add mushrooms; cook until vegetables are just tender, about 4 minutes.
Stir in rice; cook mixture, tossing to coat, for 1 minute. Stir in stock, the water, salt, pepper, cumin, turmeric, and paprika. Bring to a boil, and reduce to a simmer; cook, uncovered, until the rice is just tender and most of the liquid has evaporated, 15 to 20 minutes. Stir in the corn, black beans, and 1/2 cup grated cheese. Divide the mixture among the bell-pepper shells, mounding it. Peppers can be made a day ahead up to this point.
Place peppers in a glass baking dish, and fill with 1/4 inch water. Bake until peppers are tender and filling is hot, about 1 hour. Sprinkle with remaining 1/4 cup cheese, and cook until cheese is melted, about 1 minute more.