Make the pickled beets:
Place each beet in its own small pot, cover with water, and simmer until a paring knife can easily pierce the beets, about 30 minutes. Cool beets, peel, and slice into thin half-moons. Reserve each beet in a separate bowl to ensure that the red beet slices don’t stain the yellow ones. Meanwhile, combine the vinegar, sugar, and 1 cup water in a small pot. Bring to a boil, remove from heat, and divide the hot brine among the beets; let sit at room temperature until pickled, 3 to 4 hours.
Make the pistachios:
Preheat the oven to 300°F. In a small mixing bowl, whip the egg white until frothy, then add the pistachios and spices, toss to coat, and spread on a parchment-lined baking sheet. Bake until golden and fragrant, 15 minutes. Nuts will crisp as they cool.
Make the preserved-lemon vinaigrette:
In a medium bowl, whisk together all the ingredients until combined.
Assemble the salad:
Arrange the pickled beets on a serving platter. Toss the arugula in the preserved-lemon vinaigrette and place on top of the beets. Sprinkle with the spiced pistachios, drizzle with balsamic, and top with fresh black pepper. Serve immediately.
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