Place a medium to large pot on medium-high heat. Add butter, garlic and the green onions (except small amount of greens reserved for garnish) to the pot, and sauté until fragrant.
Add the mushrooms, and sauté, stirring regularly, until mushrooms are fully cooked. By the time they are fully cooked, they will likely have released quite a bit of liquid in the pot.
Add the white wine, oregano and dill, stirring well. Allow the mix to cook for around 5 minutes before turning the heat down to medium heat.
Add the milk and broth. Allow the soup to heat up, then remove about a cup to a separate small bowl and add the cream cheese and stir until the cream cheese has mostly melted and become more liquid. Pour this mix into the soup and stir well until fully mixed into the soup
Taste-test and add salt and pepper to prefer taste. Cook for another 5 minutes. Remove from heat, and serve immediately, with reserved green onions as garnish.