Bake beets in a single layer on a lightly greased baking sheet at 425° for 40 to 45 minutes or until tender, stirring every 15 minutes; cool completely.
Whisk together 1 cup olive oil, red wine vinegar, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a small bowl; set vinaigrette aside.
Peel beets; cut into wedges. Toss together red beets, 1/4 cup vinaigrette, 1/8 teaspoon salt, and 1/8 teaspoon pepper; set aside. In a separate bowl, toss together golden beets, 1/4 cup vinaigrette, remaining 1/8 teaspoon salt, and remaining 1/8 teaspoon pepper; set aside.
Cut top 3 inches from asparagus, cook in boiling water to cover 1 to 2 minutes or until crisp-tender. Plunge into ice water to stop the cooking process; drain. Combine asparagus and 1/2 cup vinaigrette; set aside.
Cut goat cheese into 6 equal slices. Place 1 cheese slice in a 3-inch round cutter or ring mold; sprinkle with 1/2 teaspoon chives. Press chives into cheese; remove cutter. Transfer cheese to a serving plate. Repeat procedure with remaining cheese and chives.
Arrange asparagus spears evenly on top of cheese rounds. Top with beets and drizzle with remaining vinaigrette. Sprinkle cracked pepper and olive oil around rim of plate, if desired.
*Note: Small red beets may be substituted for golden.