|Preheat oven to 350 degrees.
Make a wet rub by combining paprika, salt, pepper and 2 tablespoons olive oil. Remove 1 tablespoon of rub, combine with thyme and garlic and reserve; rub remainder of rub, minus the garlic and thyme, into exterior of pork. Rub the garlic-thyme seasoning mixture in between the two pieces of tied loin. (The garlic and thyme would burn when seared in hot oil.)
In a Dutch oven or other heavy-bottomed pan large enough to accommodate roast, heat remaining tablespoon oil over medium high heat. Brown roast on all sides, about 10-15 minutes. Remove roast from pan and reserve. Deglaze pan, continuously whisking, with 1/4 cup of beer. Stir in maple syrup and reduce heat.
Sauté onions and apples in mixture, stirring occasionally, until softened; stir in remaining beer. Center pork in the middle of apples and onions in pan or Dutch oven and cover tightly with aluminum foil and lid. Bake 1 to 1 1/2 hours, depending on size of roast, or until pork is tender, but not over-cooked. (I remove mine when it temps at 140 degrees; the temperature will rise as it sits. Pork loin roasts are leaner than other cuts, so take care not to overcook it or it will be dry.) While pork is roasting make Apple-Kraut (see below).
Remove strings from pork. Place pork loin on a serving platter and tent with foil to retain heat. To make a sauce, purée pan juices, apples and onions in a blender or food processor.
Slice pork and serve with apple-onion gravy and Apple-Kraut.
Over medium low heat, heat butter or oil. Add onions to pan and sauté with brown sugar and a pinch of kosher salt 10 minutes, stirring occasionally.
Raise heat to high, then deglaze pan with 1/4 cup of the reserved beer or apple cider. Stir, let reduce a minute, then add sliced apples to the onion mixture.
Simmer an additional 10 minutes, or until apples are tender but not falling apart. Stir in sauerkraut, caraway seeds and remaining beer or cider. (May be made a couple of hours in advance, up to this point.) Heat through before serving.