1. Rinse plums, cut in half, and pit. Cut about 1 cup thin slices, cover, and set aside up to 6 hours. Cut remaining fruit into chunks and drop into a blender or food processor; whirl until puréed.
2. In a 10- to 12-inch nonstick frying pan over high heat, boil plum purée, shallots, garlic, sugar, vinegar, port, and broth, stirring often until reduced to 1 1/2 cups, about 10 minutes. Pour into a bowl; cover and let stand at room temperature up to 6 hours. Rinse and dry pan.
3. Meanwhile, rinse pork, pat dry, and cut across the grain into 3/4-inch-thick slices. A few slices at a time, place meat between sheets of plastic wrap and, with a flat mallet, pound firmly but gently to an even 1/4 inch thick. Roll meat in plastic wrap, put in a plastic bag, seal, and chill up to 6 hours.
4. In the frying pan over high heat, combine 1 tablespoon oil and the butter. When butter is sizzling, peel plastic wrap from meat and fill pan with 1 layer of pork. As pieces shrink, add more meat. Cook, turning to brown on each side, until no longer pink in center (cut to test), about 5 minutes per batch. Add more oil as needed. As scaloppine is cooked, transfer to a platter and keep warm in a 200° oven.
5. Pour port-plum sauce into frying pan and stir over high heat until boiling; pour into a small bowl. Garnish scaloppine with reserved plum slices. Ladle plum sauce over portions; add salt and pepper to taste.