Warning: This roast chicken tastes even better than it looks.
Preheat the oven to 450°. Place the chicken, breast-side down, on a cutting board. Using kitchen shears, cut along both sides of the backbone to remove it. (Freeze it to save for making stock!) Open the chicken like a book, then flip it over, breast-side up. Place your palms over the breast and press down firmly until you feel the breastbone crack and the chicken lies flat. Tuck the wings underneath the body. Transfer the chicken, breast-side up, to a wire rack.
In a small bowl, whisk 2 tsp. oil, the mustard, and poultry seasoning. Brush all over the chicken; season with salt and pepper. Toss the cut beets with the remaining 2 tbsp. oil; season. Transfer to a rimmed baking sheet. Place the rack with the chicken on top of the beets.
Roast until the chicken is cooked through and the beets are tender, 40 to 45 minutes. Transfer the chicken to a cutting board; let rest for 10 minutes. Cut into 8 pieces.
Meanwhile, in a small bowl, whisk the crème fraîche and horseradish; season. In a large bowl, toss the beets and any juices from the baking sheet with the beet greens and orange juice; season. Transfer the chicken and beets to a platter. Drizzle the beets with the horseradish crème fraîche and sprinkle with the orange zest and tarragon.