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 Tbsp garlic mixture. Rub remaining garlic mixture under skin of chicken breasts and thighs.
Arrange potatoes and carrots around chicken and bake 20-25 minutes.
Remove pan from oven and arrange Brussels sprouts around chicken, and spread remaining 2 Tbsp garlic mixture on breasts; return to oven, and bake until a meat thermometer inserted in thickest portion registers 165°F, about 20 more minutes.
Drizzle with lemon juice, and let stand 10 minutes. Carve chicken, and serve with pan juices.