In a food processor fitted with the shredding disk, shred carrots. Transfer to a medium bowl; toss with 2 tablespoons lemon juice, 1 tablespoon oil, and half the scallions. Season with salt and pepper; set aside.
Wipe food processor clean, and fit with chopping blade. Process 1 bun until finely ground. Add lentils, egg, half the mint, 1 teaspoon salt, 3/4 teaspoon pepper, and remaining scallions. Pulse just until combined.
Dividing evenly, shape mixture into four 3 1/2-inch patties. In a large skillet, heat remaining 2 tablespoons oil over medium. Cook burgers until browned and firm, 4 to 5 minutes per side.
In a small bowl, combine yogurt, remaining mint, and remaining tablespoon lemon juice; season with salt and pepper. Place burgers on buns, and top with yogurt sauce and lettuce. Serve with carrot salad.
The food processor is such a handy tool here. Shredding carrots for the salad and blending canned lentils and other ingredients for the burgers is done in mere seconds.