Almost-Famous Chicken Sandwiches

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2 skinless, boneless chicken breasts (about 6 ounces each)

Kosher salt and freshly ground pepper

1 teaspoon paprika 

Peanut oil, for frying

1 large egg

1/2 cup nonfat milk

3/4 cup all-purpose flour

1/4 cup whole-wheat flour

1 tablespoon malted milk powder

1 tablespoon plus 1 teaspoon confectioners' sugar

1/4 teaspoon baking soda

1/4 teaspoon dry mustard

1 sour pickle, cut into 8 slices, plus 1 tablespoon pickle juice from the jar

2 teaspoons white vinegar

4 soft hamburger buns, split

1 tablespoon unsalted butter, softened