Melt butter in a large pot over medium. Add celery, carrots, and onion. Sprinkle with flour and salt. Cook, stirring frequently until carrots start to soften, about 5 min.
Add potatoes, wine, broth, milk, and herbes de Provence. Bring to a boil, then reduce heat to medium-low. Simmer, partially covered, until potatoes are tender, about 15 min. Stir in cream and trout. Continue simmering for 2 min. Ladle into bowls and top with parsley just before serving.
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