Trim beet, leaving about 1 inch of stem attached, and wrap tightly in foil. In a small baking pan roast beet in middle of oven until tender, about 1 hour.
Pick over crab meat to remove any bits of shell and cartilage. Slightly break up large pieces of crab meat and in a small bowl stir together with chives, lemon juice, 1 tablespoon sour cream, and salt and pepper to taste. Crab mixture may be made 6 hours ahead and chilled, covered.
Unwrap beet carefully. When beet is just cool enough to handle, slip off skin and stem and cut beet into a few pieces. While beet is still warm, using a mortar and pestle mash beet until smooth. In a small bowl stir together mashed beet, vinegar, remaining ‚ tablespoon sour cream, and salt and pepper to taste. Beet purée may be made 1 day ahead and chilled, covered.
Cut apple through stem end into quarters and core.
Cut apple quarters crosswise into 1/4-inch-thick slices. Top each apple slice with 1/4 teaspoon beet purée and some crab mixture.
