Place lentils and broth in a medium saucepan. Bring to a boil then reduce the heat to Low. Cover and simmer for 20 minutes, or until tender but not mushy. Drain any remaining liquid. (Skip this step if using canned lentils.)
Set a medium skillet over medium heat and add olive oil. When the oil is hot, add onion and bell pepper. Cook 3 minutes, stirring frequently, until softened.
Stir in cooked lentils, ketchup, mustard, brown sugar, garlic powder, smoked paprika, Worcestershire sauce, and black pepper.
Reduce heat to low and cook uncovered for 10 minutes, stirring occasionally, until heated through.
Adjust seasoning to taste, then spoon the mixture onto toasted burger buns. Serve warm.