How to Make It
Combine olive brine, 2 tablespoons oil, sage, lemon zest, and 1 teaspoon salt in a large ziplock plastic bag. Prick pork all over with a fork, and add to bag. Squeeze air out of bag and seal. Refrigerate 3 hours. Drain and discard marinade. Pat pork dry.
Preheat oven to 375°F. Heat 2 tablespoons oil in a large ovenproof skillet over medium-high. Season pork evenly with 1 1/2 teaspoons salt and pepper. Add pork to skillet; sear until one side is browned, about 6 minutes. Flip pork; scatter olives and garlic in skillet, and transfer to preheated oven. Roast until a thermometer inserted in thickest portion of pork registers 145°F, 10 to 13 minutes. Transfer pork to a cutting board; let rest 10 minutes.
Return skillet to stove over medium-high. Add wine, stirring and scraping to loosen browned bits from bottom of skillet. Simmer until liquid is almost evaporated, about 2 minutes. Add beans and stock. Return to a simmer over medium-high. Cook, mashing roasted garlic and some of the beans with the back of a wooden spoon, until mixture is slightly creamy, 3 to 4 minutes. Remove from heat; stir in parsley, remaining 1 tablespoon oil, and remaining 3/4 teaspoon salt. Add salt to taste if needed. Slice pork, and serve with bean mixture.
Mild green olives such as Cerignola (and their brine) can be substituted for Castelvetrano.