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Heat the oil in a large saucepan over medium heat. Add the onion and cook until softened and just beginning to brown, about 5 minutes.
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Pour in the lentils and broth. Bring the lentils to a boil, then partially cover and reduce heat to low.
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Simmer for about 20 minutes, or until most of the liquid has absorbed and the lentils are tender enough to eat, but still firm enough to hold their shape.
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Stir in the tomato sauce, molasses, vinegar, sugar, tomato paste, tamari, mustard powder, garlic, and cayenne. Increase the heat to medium and bring the mixture to a slow boil.
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Continue to cook until the sauce has thickened, 10-15 minutes more, stirring frequently.
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Divide the lentil mixture onto the hamburger buns and serve. Leftovers will keep in the freezer for 2-3 months.