Sear the Meat: Heat a large cast-iron skillet or heavy-bottomed pan over High heat. Once the pan is shimmering, place the pork shoulder in the pan. Sear for approximately 5 minutes per side until a deep golden-brown crust forms.
Create the Pocket: Transfer the seared meat to a large cutting board. Using a long, sharp carving knife, carefully cut a deep, wide pocket into the thickest part of the shoulder, being careful not to cut all the way through to the other side.
Stuff: Push the cheese cubes and bacon chunks deep into the pocket until it is tightly packed.
Seal the Opening: Insert wooden toothpicks at 1-inch intervals across the opening of the pocket. Take a length of kitchen twine (string) and lace it around the toothpicks in a crisscross "shoelace" pattern, looping it tightly to seal the edges of the meat together.
Prepare for Baking: Preheat your oven to 375°F (190°C). Line a rimmed baking sheet with parchment paper.
Season: Place the stuffed shoulder onto the baking sheet. Using a pastry brush, coat the entire surface of the pork with the 2 tablespoons of melted butter. Sprinkle evenly with the smoked paprika, salt, and pepper.
Bake: Roast in the center of the oven at 375°F for 50 minutes. The internal temperature of the meat (not the filling) should reach 145°F (63°C) for a juicy finish.
Rest and Serve: Remove from the oven and let the roast rest for 10 minutes. Carefully remove the string and toothpicks before slicing into thick rounds.
Storage: Store leftovers in an airtight container in the refrigerator for up to 4 days. Reheat slices gently in a covered pan over low heat to keep the cheese creamy.