Brown the pork: Heat 2 to 3 tablespoons olive oil in a large pot or deep skillet over medium-high heat. Add the pork shoulder and brown for about 3 to 4 minutes per side. Work in batches if needed so the meat sears instead of steaming.

Cook the onion and carrot: Add the chopped onion and cook until softened and lightly golden, about 3 to 4 minutes. Stir in the diced carrot and cook for 5 minutes, just until it begins to soften.

Add the potatoes, water, and butter: Spread the cubed potatoes evenly over the pork and vegetables without stirring. Pour in 2 cups water or broth, then place 3 tablespoons butter on top. Cover and simmer gently for about 25 minutes, until the potatoes are starting to soften.

Season and finish: Uncover and season with salt, black pepper, smoked paprika, whole peppercorns, and bay leaves. Add 3 cloves minced garlic, stir gently, cover again, and simmer for 10 to 15 minutes, until the pork is tender and the potatoes are soft but still holding their shape.

Serve: Remove the bay leaves, sprinkle with fresh parsley, and serve warm. Let the stew rest for 10 minutes if you want a slightly thicker broth.