Place pork roast into the bottom of a slow cooker and cover with sauerkraut with juice, applesauce, brown sugar, and 1 can beer. Set cooker to High, cover, and cook 3 hours. Reduce heat to Low and cook 4 more hours.

About 1 1/2 hours before serving time, heat water with reserved pork trimmings in a large saucepan over low heat and simmer to make broth, about 45 minutes. Strain broth, discard trimmings, and pour remaining can of beer and kraut juice into the simmering broth. Bring the broth to a boil.

Mix baking mix and milk in a bowl to make a sticky dough. Drop by spoonfuls into the boiling broth. Reduce heat to low, cover, and simmer dumplings for 10 minutes; uncover and simmer until dumplings are set, about 10 more minutes. For chewier dumplings, boil a few minutes longer. Remove dumplings from the broth with a slotted spoon and transfer to a serving bowl.