Using stems as a handle, slice beets paper-thin with slicer (wear protective gloves to avoid staining hands). Toss with vinegar and 1/4 teaspoon salt in a bowl, then let stand 5 minutes to wilt.
Divide beets among 4 large plates, overlapping slices slightly. Top with goat cheese.
Toss arugula and endive with 1 tablespoon oil and salt and pepper to taste, then mound over goat cheese.