Preheat the oven to 450 degrees.
On a baking sheet, toss the beets with 1 1/2 tablespoons of the oil to coat. Season with salt and pepper. Arrange the beets in the center of the baking sheet, forming a bed large enough to hold the salmon. Roast the beets for about 20 minutes or until crisp-tender.
Place the salmon on top of the beets. Brush the salmon with 1/2 tablespoon of the oil and season with salt and pepper.
In a large bowl, mix the parsley, chives and tarragon. Sprinkle all but 1 tablespoon of the mixed herbs over the salmon.
Roast the salmon for about 15 minutes, or until cooked to medium-rare doneness so it is slightly rosy in the center. Remove from the oven.
Meanwhile, whisk the remaining 4 tablespoons oil, the shallots, lemon zest and juice into the remaining mixed herbs. Season the dressing to taste with salt and pepper.
Toss the mixed greens with 2 tablespoons of the dressing. Drizzle the remaining dressing over and around the salmon and beets and serve the greens alongside.