Place the cubed pork, cold water, sugar, and salt into a food processor. Process until the meat is smooth and sticky, resembling ground pork.

Divide the processed pork into four portions, each weighing about 4 ounces. Moisten your hands with cold water to prevent sticking and form each portion into a roughly 7x3-inch rectangular patty.

Place the patties between two pieces of parchment paper on a baking sheet. Freeze until completely solid, about 2 hours, to ensure the patties hold their shape during cooking.

Preheat a griddle or cast-iron skillet over high heat until it is very hot, but not yet smoking. Place the frozen pork patties directly onto the hot griddle. Cook for about 3 to 4 minutes per side, until golden brown and cooked through (an internal temperature of 145°F for pork that you ground from a pork butt). Remove the patties from the griddle and let them cool for a few minutes.

While the patties are cooling, spread 1/2 tablespoon of softened butter on the cut side of each hoagie roll. Place the rolls cut side down on the hot griddle until lightly browned, about 1 minute.

Pour the barbecue sauce into a large, shallow bowl. Using tongs, dip each pork patty into the sauce, ensuring both sides are fully coated. Let any excess sauce drip back into the bowl.

To build the sandwiches, place a pork patty onto the bottom half of a toasted hoagie roll. Top the patty with 2 pickle slices and about 2 tablespoons of sliced onions. Cover with the top bun and serve immediately.