Combine cider, broth, vinegar, and cinnamon stick in medium saucepan; simmer over medium-high heat until liquid is reduced to 1 cup, 10 to 12 minutes. Remove cinnamon stick and discard. Set sauce aside until pork is cooked.
Put 1 tablespoon butter in a clean skillet and heat over medium heat until melted and foaming subsides. Add shallot and apple and cook, stirring occasionally, until softened and beginning to brown, 1 to 2 minutes. Remove skillet from heat and add Calvados. Return skillet to heat and cook about 1 minute, scraping bottom with wooden spoon to loosen browned bits.
Add reduced cider mixture, any juices from resting meat, and thyme; increase heat to medium-high and simmer until thickened and reduced to 1 1/4 cups, 3 to 4 minutes. Off heat, whisk in remaining 3 tablespoons butter, and adjust seasonings with salt and pepper. Pour sauce over pork and serve immediately.