Heat the oven to 300 degrees F. In a medium bowl, stir together 2 cups grated beets, the lemon zest, and the chili flakes, and season with salt and pepper. Line a baking sheet with parchment paper, then place the salmon on the sheet.
Season the salmon all over with salt and pepper, then cover the top and sides of the salmon with the beet mixture. Set aside at room temperature for 30 minutes to marinade.
Meanwhile, make the salad: In a medium bowl, combine the the grated carrot, lemon juice, and remaining 1½ cups grated beets. Season with salt and pepper and set aside (the vegetables will soften as they sit).
After 30 minutes, scrape the beets off the salmon, and discard or compost. On a baking sheet, roast the salmon until cooked through, or until it reaches 120 degrees F in the thickest part, 15 to 20 minutes.
While the salmon cooks, make the labneh: In a liquid measuring cup, stir together the saffron, chili flakes, and 2 tablespoons boiling water. Let sit until the water is bright orange. Pour the infused water over the labneh, stir to combine, and season with salt and pepper to taste.
Garnish the salmon with chives, flaky salt, black pepper, and chili flakes as desired. Serve alongside the beet salad and labneh.