Melt butter in a large saucepan over medium-low heat. Cook and stir leek, shallots, and 1/2 teaspoon salt in the melted butter until softened, 10 to 15 minutes.
Stir wine into leek mixture, increase heat to medium, and cook for 2 minutes. Add chicken broth and bring to a simmer.
Mix fennel and potatoes into leek mixture and simmer, stirring occasionally, until potatoes are nearly tender, about 10 minutes. Season with salt, black pepper, and cayenne pepper. Add cream and stir to combine.
Stir fish and tarragon into soup, cover and cook for 3 minutes. Stir gently, reduce heat to medium-low and cook until fish flakes easily with a fork, about 5 minutes. Season with salt and black pepper.
