Season pork generously with salt all over. Let sit at room temperature at least 1 hour, or preferably chill overnight, then let sit at room temperature 1 hour before cooking.

Preheat oven to 300°. Heat oil in a large Dutch oven or heavy wide saucepan over medium-high. Add garlic, cut sides down, and cook until golden brown, about 1 minute. Transfer garlic to a plate. Add pork to pot and cook, undisturbed, until golden brown, 3–5 minutes. Turn and cook, reducing heat to medium if needed so it doesn't burn, until golden brown on all sides, another 3–5 minutes.

Return pork and garlic to pot. Add milk, sage, chiles, and lemon peel. Transfer pot to oven and bake, uncovered, turning pork every 30 minutes, until meat is very tender and can easily be pierced with a paring knife and milk has reduced and formed curds, 2½–3 hours. Transfer meat to a cutting board and let rest 15 minutes.

Using a spoon, skim off most of the fat from the braising liquid; discard sage and lemon peel. Taste sauce and season with salt, if desired.

Slice pork and transfer to a platter. Arrange roasted garlic alongside pork, if desired. Spoon sauce over. Top with parsley; season with pepper.