In a medium size frying pan, heat 1 tablespoon of olive oil over medium heat. Add onion and carrot and cook until they begin to soften, about 3 minutes. Stir in red pepper flakes and lentils, and cook for 1 minute. Stir in broth, water and salt, and bring to a boil. Reduce heat to medium-low and simmer, uncovered, until lentils are just tender and liquid is absorbed, 22 to 24 minutes. Stir in spinach during last min of cooking.
Remove pan from heat and let cool for 10 minutes. Stir in bread crumbs and egg. (Add up to 2 tablespoon water if mixture is to dry.)
For the sauce, stir yogurt, feta, garlic and dill together in a small bowl. Set aside.
Heat the remaining 1 tablespoon of oil in a large pan over medium heat. Scoop heaping 1/2-cup portions of lentil mixture and form into 1-inch-thick patties. Add to pan and cook until golden, about 3 to 4 min. Flip and continue cooking, 3 minutes.
Serve on burgers buns topped with sauce and add your favorite toppings!
