Melt butter in a large pot over medium heat. Add diced onion and cook until soft and translucent, about 4 minutes.
Add sliced mushrooms and cook until golden brown and liquid has evaporated, about 6 to 8 minutes. Stir in paprika, dill, and soy sauce.
Sprinkle flour over the mushrooms and stir to coat evenly. Pour in vegetable broth and milk, stirring constantly to prevent lumps. Bring to a gentle simmer.
Reduce heat to low and simmer for 15 minutes, stirring occasionally, until the soup thickens to a creamy consistency. Season with salt and pepper.
Remove from heat and stir in sour cream and lemon juice. Taste and adjust seasoning. Serve hot, garnished with fresh dill.