Preheat the oven to 425 degrees F. Using a mister or pastry brush dipped in olive oil, coat a 17 x 12-inch metal baking sheet with sides (or two smaller pans with sides).
Place the drumsticks on the pan. Season on all sides with 1/4 teaspoon salt.
Bake for 20 minutes.
Remove the pan from the oven.
Add the carrots, scallions, parsley, lemon zest, 3/4 teaspoon of salt, and 2 tablespoons of oil. With a pancake turner, toss thoroughly to combine the vegetables and seasonings.
Spread in a single layer.
Bake for about 25 minutes, turning occasionally, or until no pink remains at the chicken bone and the carrots are tender.
Meanwhile, place a covered large pot of water over high heat. When the water boils, add 2 teaspoons of salt and the fettuccine. Stir. Cover and return to a boil. Uncover and boil, stirring occasionally, for about 3 minutes, or until tender.
Drain and return to the pot.
If the chicken skin is not well browned, heat the oven broiler. Broil the chicken 6 inches from the heat source for about 3 minutes, or until browned.
Remove the drumsticks to a platter.
Add the carrots, scallions, and pan juices to the fettuccine. Toss to combine.
