In a large mixing bowl, combine the finely chopped corned beef, thoroughly drained sauerkraut, shredded Swiss cheese, and softened cream cheese until it forms a cohesive dough.

Scoop out tablespoon-sized portions and roll them tightly into uniform spheres. Place them on a parchment-lined baking sheet and chill in the refrigerator for 30 minutes.

Place the all-purpose flour, beaten eggs, and panko breadcrumbs into three separate shallow bowls.

Roll each chilled sphere in the flour, dip it completely into the egg wash, and then press it firmly into the panko to create a heavily textured coating.

Heat 4 cups of oil in a heavy Dutch oven to 350°F (175°C). Fry the breaded spheres in small batches for 3 to 4 minutes until they are crispy and golden-brown. Drain on a wire cooling rack.

Whisk together the mayonnaise, ketchup, and green relish in a small bowl. Stir in the red chili flakes and cracked black pepper until evenly speckled.

Transfer the dipping sauce to a small white ramekin. Serve immediately with the hot fried bites, scattering fresh green parsley leaves over the platter.