Bring a large pot of water to a boil. Add the udon noodles and cook according to package instructions, usually 2-3 minutes for fresh or frozen noodles. Once cooked, drain and rinse under cold water to stop cooking and set aside.
Heat the vegetable oil in a large pot over medium heat. Add the minced garlic and grated ginger, stirring frequently until fragrant, about 1-2 minutes. Add the sliced mushrooms and sauté until they release their moisture and become tender, about 5-7 minutes.
Pour in the vegetable broth and bring to a gentle simmer. In a small bowl, whisk the miso paste with a few tablespoons of hot broth until smooth, then stir this mixture back into the pot. Add soy sauce and adjust seasoning to taste.
Add the fresh spinach to the simmering soup, stirring until wilted, about 1-2 minutes. Finally, add the cooked udon noodles to the pot, heating everything through for another 2 minutes.
Ladle the soup into bowls. Garnish generously with sliced green onions and toasted sesame seeds. Serve immediately while hot and comforting.