Prepare the Eggs: Preheat the oven to 300°F. In a saucepan, add 2 eggs and cover them with cold water. Bring to a boil, then cover the pan and turn off the heat. Allow the eggs to sit for 5 minutes before transferring them to cold water. Once cooled, peel and slice the eggs in half lengthwise.

Mix the Embutido Filling: In a large mixing bowl, beat the remaining 2 eggs. Add the grated Edam cheese, diced smoked ham, peas, raisins, minced onion, mashed Vienna sausages, relish, ketchup, salt, and black pepper. Gently fold in the ground pork and mix by hand until thoroughly combined, ensuring the mixture holds together. To test the seasoning, cook a small spoonful in a frying pan over medium heat and adjust with more salt or pepper as needed.

Form the Embutido Roll: Line a large baking pan with greased foil, leaving enough overhang on all sides. Spread the embutido mixture into a rectangular shape, approximately 12 by 17 inches. Arrange the boiled egg halves in a line along one of the long edges of the rectangle. Carefully roll the meat mixture, starting from the egg side, using the foil to assist in creating a tight cylinder. Once rolled, pat the embutido to form a uniform shape and seal the foil tightly, crimping the ends to secure it.

Bake and Broil: Place the foil-wrapped embutido in the center of the pan and bake for 45 minutes. After this time, remove the embutido from the oven and turn on the broiler. Unwrap the foil, allowing the embutido to rest for 10 minutes before broiling it for 3 minutes or until lightly browned. Let the embutido rest for another 3 minutes, then slice it into serving pieces.

Prepare the Sauce: While the embutido is baking, make the sauce by combining 3 tablespoons of calamansi juice, soy sauce, and chicken broth in a small saucepan. Simmer until the sauce has reduced by half. In a separate bowl, mix the remaining calamansi juice with cornstarch and whisk this into the sauce. Bring the mixture to a boil, then remove it from heat once thickened. Reheat the sauce when the embutido is done, adding butter one chunk at a time until fully incorporated.