How to Make It
Preheat oven to 450°F. Rinse chicken, and pat dry. Place chicken, breast side down, on a cutting board. Using poultry shears, cut along both sides of backbone, and remove backbone. (Discard or reserve for stock.) Turn chicken breast side up, and open the underside of chicken like a book. Using the heel of your hand, press firmly against breastbone until it cracks. Place chicken in a large rimmed baking pan. Tuck wing tips under chicken so they don't burn.
Combine garlic and salt on a cutting board. Using the flat edge of a knife, mash into a paste. Combine garlic paste, butter, thyme, zest, and pepper in a bowl. Set aside 2 tablespoons of the garlic mixture. Rub remaining garlic mixture under skin of chicken breasts and thighs.
Bake chicken in preheated oven 10 minutes. Remove pan from oven. Reduce heat to 400°F. Arrange potatoes and carrots around chicken; return to oven, and bake 20 minutes. Arrange Brussels sprouts around chicken, and spread remaining 2 tablespoons garlic mixture on breasts; return to oven, and bake until a meat thermometer inserted in thickest portion registers 165°F, about 20 minutes. Drizzle with lemon juice, and let stand 10 minutes. Carve chicken, and serve with pan juices.