Melt the butter in a large pot over medium heat. Cook and stir the onions in the butter until fragrant, about 5 minutes. Add the mushrooms and continue cooking until the mushrooms are tender, about 5 minutes more. Stir the dill, paprika, soy sauce, and chicken broth into the mushroom mixture; reduce heat to low, cover, and simmer 15 minutes.
Whisk the milk and flour together in a small bowl. Stir the mixture into the soup. Add the tomato and Hungarian wax pepper. Return cover to the pot and simmer another 15 minutes, stirring occasionally. Season with salt and pepper. Mix the sour cream into the soup and continue cooking and stirring until the soup has thickened, 5 to 10 minutes more. Remove the Hungarian wax pepper and tomato and discard before serving the soup.
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