Combine first 6 ingredients in a medium saucepan; bring to a boil. Cook until reduced to 1 1/2 cups (about 10 minutes). Discard cinnamon sticks.
Heat oil in a large nonstick skillet over medium-high heat. Add pork; sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper. Cook 6 minutes on each side or until pork is done. Transfer pork to a platter. Add broth mixture to pan, scraping pan to loosen browned bits. Add persimmon and cranberries; cook 5 minutes or until cranberries begin to pop, stirring occasionally. Add remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, and butter, stirring until butter melts. Serve with pork.