Preheat the oven to 375˚ F. Microwave 3 tablespoons butter in a microwave-safe bowl until melted, about 1 minute. Toss with the bread cubes on a baking sheet; spread in a single layer. Bake, stirring occasionally, until crisp, 20 to 25 minutes.
Meanwhile, melt the remaining 3 tablespoons butter in a large pot over medium-high heat. Add the leek and cook, stirring, until softened, 2 to 3 minutes. Add the garlic and cook until softened, 1 to 2 minutes. Add the potatoes, chicken broth, 1 cup pickles and 2 cups water; cover and bring to a boil. Reduce the heat to a simmer, cover and cook until the potatoes are tender, 10 to 12 minutes. Stir in the chicken and cook until heated through, 2 to 3 minutes.
Stir the dill and 1 tablespoon pickle brine into the soup; season with salt and pepper. Add more pickle brine to taste. Divide the soup among bowls and top with the croutons, the remaining 1/4 cup pickles and sour cream.