Place the cut chicken pieces in a large bowl and add the milk and dill pickle juice; cover and refrigerate for at least 20 minutes up to overnight.
Remove the marinated chicken from the fridge and add the egg to the mixture, stirring to combine.
In a large zip-top bag or shallow bowl, combine the flour, powdered sugar, paprika (if using), salt, and pepper until evenly mixed.
Drain the chicken pieces from the marinade (discard excess liquid), then add them to the flour mixture and shake or toss until fully coated.
Heat about 1–2 cups of peanut or canola oil in a cast iron or heavy skillet over medium-high heat until shimmering and hot but not smoking.
Working in batches to avoid crowding, fry the coated chicken pieces until golden brown and cooked through, about 3–5 minutes per batch, turning as needed.
Transfer cooked nuggets to a cooling rack set over a tray or to a plate lined with paper towels to drain while you finish the remaining batches.
Whisk together the mayonnaise, yellow mustard, honey, and lemon juice (or vinegar) in a small bowl; season with salt and pepper to taste and chill until serving.
Serve the nuggets immediately with the honey mustard dipping sauce.