Cover and bring a medium saucepan of salted water to a boil. Prepare an ice-water bath. Add peas, and cook, uncovered, until tender but still bright green, about 3 minutes. Remove peas, drain, and then plunge peas into ice water to stop cooking. Drain well. Add beets to boiling water, and cook until tender, about 40 minutes. Remove from heat, drain, and then plunge beets into ice water to stop cooking. Drain well. Peel beets, and cut into 1/8-inch pieces. Transfer to a large bowl.
In a deep container, combine chives and vinegar using an immersion blender. Add oil in a slow, steady stream, and blend until emulsified (alternately, the vinaigrette can be made in a regular blender). Season with salt and pepper. Strain mixture through a fine sieve.
In a medium bowl, combine peas and cucumbers. Add 1/4 cup chive oil, and mix to combine. Gently toss in avocados.
To serve, place a 9-inch ring mold on a platter. Fill with the beet mixture. Top with pea mixture, spreading evenly. Drizzle with a little of the remaining chive oil (reserve any extra for another use). Top with horseradish cream. Remove ring mold, and serve.