Rinse the chicken under cool water, then pat dry with paper towels. Place it breast-side down on a cutting board. Using kitchen shears, cut along both sides of the backbone to remove it. Flip the chicken breast-side up, press down on the breast to flatten it.
Gently slide your fingers between the skin and meat over the breast, thighs, and legs to create a pocket for the seasoning. Be careful not to tear the skin.
Rub about 3 teaspoons of salt under the loosened skin, spreading it evenly over the breast, thighs, and legs. Flip the chicken over and sprinkle another ½ teaspoon of salt on the underside. Place the chicken breast-side up on a baking sheet, cover, and refrigerate for 20 minutes to let the salt absorb.
In a small bowl, combine the softened butter, rosemary, thyme, garlic, lemon zest, ¼ teaspoon salt, and ¼ teaspoon pepper. Mix until well blended.
Take the chicken out of the fridge about 25 minutes before baking to let it warm up slightly. Preheat the oven to 425°F.
Pat the chicken dry again with paper towels. Spread the herb butter mixture under the skin of the breast, thighs, and legs, reserving a little to rub on the outside of the chicken.
Arrange the sliced carrots and potatoes on a large baking sheet. Drizzle with olive oil, sprinkle with ½ teaspoon salt and ¼ teaspoon pepper, and toss to coat.
Place the chicken in the center of the pan, with the vegetables around it and roast in the preheated oven for 50-55 minutes, or until the chicken reaches an internal temperature of 165°F. Rotate the pan halfway through cooking for even browning.
Remove the chicken from the oven and let it rest for 5-10 minutes before carving. Serve with the roasted vegetables.