Place the cut chicken pieces in a large bowl and add the 1/2 cup milk and 1/2 cup dill pickle juice; cover and refrigerate for at least 20 minutes up to overnight.
Remove the marinated chicken from the fridge, discard excess liquid, then add the large egg and mix to combine.
In a zip-top bag or a shallow bowl, combine the all-purpose flour, confectioners' sugar, paprika (if using), 1 teaspoon salt, and 1 teaspoon pepper; shake or whisk to mix evenly.
Coat the chicken pieces by adding them (without excess liquid) to the bag and shaking until evenly coated, or dredge pieces in the flour mixture using your hands or a fork.
Heat 1–2 cups of peanut or canola oil in a large skillet or cast-iron pan over medium-high heat until shimmering and hot but not smoking.
Working in batches so the pan isn’t crowded, fry the coated chicken pieces 3–5 minutes per side until golden brown and cooked through (internal temperature 165°F/74°C), then transfer to a cooling rack or paper towels to drain.
Whisk together the mayonnaise, yellow mustard, honey, and lemon juice or vinegar in a small bowl to make the honey mustard; chill until serving.
Season the finished nuggets with salt and pepper to taste and serve immediately with the honey mustard.