Add the lentils to a large pot. Place the whole potato, onion, and carrots on top. Cover with water, season lightly with salt, and cook until everything is soft (about 25–30 minutes).
Drain excess water if there is any. Mash the lentils, carrots, onion, and potato together in the pot until mostly smooth but still a little chunky.
Transfer the mash to a bowl. Add breadcrumbs, chopped parsley, paprika, cumin, salt, pepper, and chili flakes (optional). Mix until the mixture holds together. Add more breadcrumbs if needed.
Form oval or square patties by pressing the mixture firmly with your hands.
Heat oil in a pan over medium heat. Fry patties on both sides until golden brown and crispy.
Slice the bread, add the sauce, tomato slices, onion-parsley mix, 3–4 lentil patties, and grilled green pepper. Press lightly and enjoy warm!