Prepare oven and chicken: Preheat oven to 450°F. Rinse chicken, and pat dry.
Remove backbone: 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.)
Press against breastbone: 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 on baking sheet: Place chicken in a large rimmed baking pan. Tuck wing tips under chicken so they don't burn.
Make garlic-herb butter: 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.
Roast chicken: Bake chicken in preheated oven 10 minutes.
Add vegetables to baking pan: 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.
Add lemon juice, and serve: Drizzle with lemon juice, and let stand 10 minutes. Carve chicken, and serve with pan juices.