To start the sauce: Heat a 2 1/2-quart nonreactive saucepan over medium heat and, when hot, add the oil.
When the oil is hot, stir in the onion, green and red pepper, celery and garlic. Cook until the vegetables are softened and fragrant, 4 to 5 minutes.
Stir in the tomato puree, tomato paste, ketchup, vinegar, molasses and Worcestershire sauce. Bring the mixture to a simmer, reduce heat to very low and simmer with the cover ajar for 1 hour.
To sauté the mushrooms: Heat an 8 - 10 inch skillet over high heat and, when hot, add the olive oil. When the oil is hot, add the chopped mushrooms and cook, stirring frequently, until the mushrooms are golden, 2 to 3 minutes. Remove from the heat and set aside.
To finish the sauce: After the sauce has simmered 1 hour, add the sautéed mushrooms, chopped plum tomatoes, oregano, “Better Than Bouillon” seasoning and some freshly ground black pepper. Return to a simmer and cook with the cover ajar for 30 minutes more. (Refer to Note below).
To brown ground meat: Heat a 10-inch non-reactive deep-sided skillet over medium heat. When hot, add the ground meat and break it up with a wooden spatula. Cook the meat until separated and no longer pink, about 4 minutes. Then remove from the stove and drain out any excess fat from skillet. Return skillet to the stove, over low heat. Stir in the sauce and bring the mixture to a gentle simmer, uncovered. Cook gently until the flavors mingle and the mixture is piping hot throughout, about 10 minutes. Add some more freshly ground black pepper and serve immediately.
To assemble Sloppy Joes and serve: While the sauce is simmering, open the hamburger buns and spread both opened sides lightly with butter. Lay the buns, buttered sides up, on a shallow baking sheet. Broil the buns until the buttered sides are nicely toasted. Spoon the ground beef mixture lavishly over buns and serve hot.
NOTE: If serving for 8 [could serve 10 for skimpier sandwiches], you will need 2 cups of sauce and 2 lbs of ground beef. If serving 4, you will need 1 cup of sauce and 1 lb. of ground beef. The remaining sauce can be frozen for later use and ease. Freeze cooled sauce in tightly sealed containers.