Heat the sesame oil in a large pot over medium heat. Add the diced onion and cook until it softens, about 4 minutes.
Stir in the minced garlic and grated ginger. Cook for 1 minute until fragrant.
Add the sliced mushrooms and cook until they release their liquid and start to brown, about 5 minutes.
Pour in the rinsed lentils and vegetable broth. Bring the mixture to a boil, then reduce to a simmer. Cover and cook for 25 minutes until the lentils are tender.
In a small bowl, whisk the miso paste with a ladle of the hot broth until smooth. Stir this mixture back into the pot along with the soy sauce. Simmer uncovered for 5 more minutes.
Ladle the stew into bowls and top with sliced green onions and sesame seeds.