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In a large saucepan or skillet, saute ground beef in 1 tablespoon of olive oil over medium heat, breaking it up as it cooks. Once the meat has browned, add another tablespoon of olive oil and add the carrots, celery, bell peppers, onion and garlic. Cook, stirring occasionally, until the vegetables are beginning to soften, about 8-10 minutes.
Add brown sugar, vinegar, Worcestershire sauce, tomato sauce, ketchup, water and paprika to the meat and vegetable mixture. Stir to combine. Simmer over medium-low heat, partially covered, stirring occasionally. After about 15 minutes, the sauce should be thickened and the vegetables should be tender. If desired, you can add a splash of water, as necessary, to thin out the sauce.
Season to taste with salt and pepper. Scoop the sloppy joes into split buns and serve.