Trim fat from pork. Heat oil and butter in a large Dutch oven over medium-high heat. Add pork, and cook 6 minutes, browning on all sides. Remove pork from pan. Set aside, and keep warm. Add onion to pan, and sauté over medium heat for 7 minutes or until browned. Set aside.
While onion cooks, prepare the milk mixture. Combine milk and bay leaves in a large heavy saucepan. Cook mixture over medium-high heat to 180° or until tiny bubbles form around edge (do not boil). Remove from heat.
Rub pork with thyme, salt, pepper, and garlic; insert meat thermometer into thickest portion of pork. Return pork to pan; add milk mixture. Cover; bring to a simmer over medium heat. Reduce heat to low; cook 1 hour and 15 minutes or until thermometer registers 160° (slightly pink), turning once. (Do not boil.) Discard bay leaves. Remove pork from pan, reserving 2 1/4 cups braising liquid. Discard remaining braising liquid.
While pork cooks, prepare the apple mixture. Combine apples and water in a large saucepan. Cover; cook over medium heat 15 minutes or until tender, stirring occasionally. Partially mash apples with a fork. Set aside; keep warm.
Thinly slice pork; serve with reserved braising liquid and apple mixture. Garnish with arugula and thyme sprigs, if desired.