Heat sesame oil in a stockpot or Dutch oven over medium-low. Once warm, add garlic, ginger, and chili flakes. Cook 5 minutes, stirring occasionally, until aromatic. Transfer garlic-chili oil to a bowl.
Add vegetable oil to pot and increase heat to medium-high. Add mushrooms; cook 6 to 7 minutes, stirring occasionally, until golden. Roughly chop two thirds of green onions and add to pan (reserve remaining for garnish); cook 2 minutes. Stir in rice vinegar, and scrape up any browned bits stuck to the bottom of pan.
Add soy sauce, fish sauce, and broth; bring to a boil. Add noodles, reduce heat, and simmer, uncovered, until noodles are cooked, about 3 to 5 minutes. Stir in reserved garlic-chili oil and edamame.
Divide into four bowls and add remaining thinly sliced green onion, plus other garnishes of choice.