In a large saucepan, cover the lentils with cold water by 2 inches and bring to a boil over high heat. Reduce the heat to moderate and simmer the lentils until they are very tender, about 10 minutes. Drain the lentils well.
Preheat the broiler. In a medium skillet, heat 2 tablespoons of the olive oil. Add the onion and cook over moderately high heat, stirring frequently, until golden, about 6 minutes. Add the carrots and two-thirds of the garlic and cook until the carrots begin to soften, about 4 minutes. Stir in the cumin and cayenne and remove from the heat. Mix in the bread crumbs, 3 tablespoons of the parsley, the eggs, salt and pepper. Form the mixture into sixteen 1/4-cup burgers.
Line a baking sheet with foil. Brush the foil and both sides of the burgers with the remaining 1/4 cup of olive oil. Broil until golden, about 3 minutes per side.
Meanwhile, in a small bowl, mix the yogurt with the lemon juice and the remaining garlic and 1 tablespoon of parsley. Serve the burgers hot, with the yogurt sauce on the side.