Bring a small pot of water for the eggs to a boil.
While water is coming to a boil, place a large heavy-bottomed pot or wok, heat cooking oil over medium heat.
When oil is hot, add shiitake mushrooms and cook until deep golden brown, 5 to 7 minutes. Add soy sauce and garlic continue cooking until mushrooms absorb all of the soy sauce. Set aside.
To heated pot, add 2 Tablespoons of coconut milk (use the extra creamy, top part from the can) and green curry paste. Saute until fragrant, about 2 minutes.
Add in vegetable stock, remaining coconut milk, brown sugar, and fish sauce. Bring to a simmer.
When broth is simmering, stir in ramen noodles. Simmer until ramen is just tender (check the package for recommended time; don’t overcook).
Meanwhile, return to the pot for the eggs and reduce to a low simmer. Gently lower eggs into simmering water. Simmer for 7 minutes if eggs are at room temperature and 8 minutes if eggs have been refrigerated. (Be sure to set a timer right after you add the eggs to the water so they don’t overcook.)
Remove ramen from heat and stir in spinach and edamame until spinach wilts.
Divide ramen and broth between large serving bowls. Top each bowl with mushrooms.
Peel eggs and slice in half. Arrange eggs on top of ramen. Serve with green onions, sesame seeds, and Sriracha.