Slice beets paper thin to 1/8-inch thick with a mandolin or very sharp knife.
Combine sugar, vinegar, ginger, and 1/3 cup water in a nonreactive saucepan. Bring to a boil, reduce heat, and simmer 3 to 5 minutes. Remove ginger.
Add beets to vinegar mixture, and bring to a boil. Cook 30 seconds (for paper thin slices) or 3 minutes (for 1/8-inch slices) or until beets are tender but still hold their shape. Remove pan from heat; allow beets to cool to room temperature in the liquid. Drain beets, reserving liquid. Place beets into a jar; cover with 1/4 cup cooking liquid.
Bring remaining cooking liquid to a boil; reduce heat, and simmer 10 minutes or until mixture thickens and reduces to 1/2 cup. Cool; cover and chill until ready to serve.
Place beets on a serving platter in a single layer. Top with cheese and arugula. Whisk together chilled liquid, olive oil, and salt and pepper to taste. Drizzle beets with dressing to taste.