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To make the tzitziki, cut the cucumbers in half lengthwise. Take a spoon and scoop out the seed pulp in the middle, throw the seed mixture away. Grate the cucumber halves on the large hole setting, 1/4 inch in diameter, or a cheese grater or food processor.
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Set out two medium sized bowls. Grab a handful of the grated cucumbers and press your hands together over one of the bowls, squeezing out as much of the water as possible. Place the handful of dry cucumbers in the second bowl. Repeat this process until all of the grated cucumber has been drained of excess water. If you want a refreshing drink, you can combine the cucumber juice with a bit of chilled water, otherwise discard the juice from the cucumber.
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Empty the yogurt into a large bowl. Mix in the shredded cucumber, garlic, olive oil, white vinegar, salt, and black pepper with a large wooden spoon. Stir until well blended. Cover and store in the refrigerator.
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To make the lamb shanks, give them a good rubbing with1 tablespoon of the olive oil, 1/2 teaspoon of the salt, 1/2 teaspoon of the pepper, and 1/2 teaspoon of the oregano. Place in a casserole pan, cover, and place in the refrigerator to marinate for 2 hours. Preheat the oven to 400 degrees Fahrenheit. Remove the pan from the refrigerator, remove the covering, and mix in the remaining ingredients. Place in the oven and cook for 2 and 1/2 to 3 hours, flipping the shank over halfway through and basting it every 45 minutes. Remove from oven and allow to cool until safe to handle. Carve the meat from the bone and cut it into 1-inch cubes. Place in a bowl for serving.
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To make the ground lamb, heat the onions in the olive oil in a large frying pan over medium heat until they begin to look transparent, about 10-15 minutes. Add the remaining ingredients and cook until the lamb meat is browned, breaking it apart with the end of your spoon. Once cooked, remove from heat and cover to keep warm.
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To make the rice, heat the water, rice, lemon juice, oregano, oil, and salt in a large thick-bottomed saucepan over high heat until boiling. Once it boils, reduce the heat to a simmer cover, and allow to simmer for 12-15 minutes or until the rice is fully cooked. Remove the pot from the heat and, keeping it covered, allow it to rest for 10 minutes before fluffing with a fork and serving.
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To assemble the burrito, place a smear of guacamole on the tortilla, add the desired amount of rice, add the desired amount of lamb, add a couple tablespoons of tzitziki, and top with queso fresco, a couple tomato slices, and a pinch of cayenne pepper. Wrap up, and serve immediately.