Pre-heat the grill to 350 degrees.
Boil the broth in a medium sized saucepan over high heat. Stir in the lentil's and 1 tsp of salt; reduce the heat to low. Cover and simmer for about 30 minutes or until lentils are tender and water is absorbed. Once cooked allow to cool to room temperature.
In a small bowl combine the flax seed and the boiling water; set aside for about 10 minutes.
Place the lentils, carrot, and mushrooms into a food processor; pulse until finely chopped but not smooth. Transfer mix to a large bowl along with the Panko, onion, garlic, thyme, and flax seed mixture. Refrigerate, covered, 2 to 3 hours.
Shape lentil mixture into 4 (1/2 inch thick) patties. Grill for about 5-6 minutes on each side until browned on both sides.
Make the sauce by combining the mayonnaise, dill, remaining 1/4 tsp of salt, and black pepper. Serve burgers on the rolls with lettuce, tomato, and a couple of tablespoons of sauce. Enjoy!