Place the potatoes and carrots in a large pot with the chicken broth and butter, then bring the liquid to a boil and cook until the potatoes soften throughout. A fork should enter the center of a large potato quarter easily, and the butter should have melted into the broth.

Stir the diced pickles into the boiling vegetable mixture and maintain steady bubbling. The pickle pieces should spread evenly among the softened carrots and potatoes.

Use a medium bowl to blend the flour, sour cream, and water into a uniform paste. The mixture should look smooth and thick, with no visible pockets of dry flour.

Whisk the paste into the bubbling soup in 2-tablespoon portions, dispersing each addition before adding the next and avoiding a violent rolling boil. Small flour beads should dissolve as you whisk, while some potato pieces break down and give the broth more body.

Pour in the pickle juice gradually, stir in the Old Bay, salt, coarsely ground pepper, and cayenne, and cook for 5 more minutes at a controlled bubble. The broth should become evenly creamy, with fragrant steam carrying both pickle tang and savory spice.

Take the pot off the heat and serve the soup immediately. Each bowl should contain tender vegetables in a creamy broth with evenly distributed pickle pieces.