Heat large non-stick skillet over medium-high heat. When pan is warm, add 3 tablespoons unsalted butter. Let butter melt completely, swirling and tilting skillet occasionally to spread butter around entire surface.
When butter is melted, add ¾ cup panko breadcrumbs. Stir to incorporate breadcrumbs into melted butter, then reduce heat under skillet to medium. Cook breadcrumbs, stirring occasionally, until butter is completely absorbed and breadcrumbs are rich golden-brown color. Be careful not to burn breadcrumbs.
Once breadcrumbs are toasted, remove skillet from heat and set aside.
Add ½ cup full-fat mayonnaise, 1 8-ounce block full-fat cream cheese, 1 cup full-fat sour cream, 3 tablespoons pickle juice, 2 tablespoons Nashville Hot seasoning, 1 tablespoon grated fresh garlic, kosher salt, and ¼ teaspoon freshly ground black pepper to large mixing bowl.
Mix ingredients together on low speed 20 to 30 seconds, then increase mixer speed to medium. Continue mixing until ingredients are fully combined and mixture is smooth, then set mixer aside.
Add ⅔ cup finely chopped dill pickles, 1 cup cooked, diced chicken, and 1 tablespoon finely snipped fresh chives to bowl. Add approximately 75% of toasted breadcrumbs to bowl and use spatula to gently fold all ingredients into cream cheese mixture.
Once all ingredients have been incorporated, transfer dip to serving bowl. Garnish dip with additional finely chopped dill pickles, finely snipped fresh chives, and remaining toasted breadcrumbs. Serve dip immediately with potato chips, celery sticks, pretzel crisps, and/or crostini.