Step 1Bring a pot of water to a boil. Trim the beets, reserving the greens, and scrub. Add them to the pot and simmer until tender, 30 to 40 minutes.
Step 2While the beets cook, saute the ginger and shallot in 1 tablespoon olive oil until softened, about 5 minutes. Toast the walnuts in a small dry skillet over medium heat until just crisp and aromatic. Set aside.
Step 3Remove the cooked beets from the water with a slotted spoon and run them under cold water to cool, leaving the water in the pot simmering. Slip off the beet skins and cut the beets into medium dice. Place them in a large bowl and add the shallot-ginger mixture, another tablespoon of olive oil, 1 tablespoon vinegar, the orange juice and salt and pepper to taste. Toss to mix.
Step 4Boil the greens in the beet water until just wilted, 3 to 4 minutes. Drain well in a colander, then squeeze out as much water as possible. Coarsely chop and place in a shallow salad bowl.
Step 5Drizzle the greens with the remaining olive oil and vinegar. Spoon the beet mixture on top and scatter the trout all around the beets. Sprinkle the walnuts over the salad.