Heat the oil in a skillet over high heat; cook and stir the onion in the oil until caramelized, 5 to 7 minutes; set aside.
Bring water to a boil in a pan over medium heat; add the lentils. Return mixture to a boil, reduce heat to low; simmer until the lentils get soupy, about 15 minutes. Stir in the bulgur, salt, and pepper; simmer until water is completely absorbed, about 10 minutes; stir in the caramelized onions. Remove from heat, cover, and allow to rest for 3 to 5 minutes.
Divide lentils into 4 even portions; squeeze one lime wedge over each portion and top with green onions and yogurt.