Instructions

Adjust oven rack to lower-middle position and heat oven to 425 degrees. Line rimmed baking sheet with aluminum foil. Using fork, mash ⅓ cup roasted garlic, 6 tablespoons softened unsalted butter, 1 tablespoon minced fresh thyme, and ¼ teaspoon table salt in bowl until smooth, about 30 seconds; set aside.
Place 4 small sweet potatoes, each lightly pricked with fork in 3 places, on large plate and microwave until potatoes yield to gentle pressure and centers register 200 degrees, 6 to 9 minutes, flipping potatoes every 3 minutes. Using tongs, transfer potatoes to half of prepared sheet. 
With chicken breast side down, use kitchen shears to cut through bones on either side of backbone of 1 (3½- to 4‐pound) whole chicken; discard backbone. Trim away excess fat and skin around neck. Flip chicken over and press on breastbone to flatten. Pat chicken dry with paper towels and, using your fingers, carefully loosen skin covering breast and legs. Rub butter mixture under skin, then sprinkle exterior of chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Transfer chicken to empty half of sheet with potatoes. Roast chicken and potatoes until breast registers 160 degrees and thighs register 175 degrees, 45 to 60 minutes, rotating sheet halfway through roasting. Transfer chicken and potatoes to carving board and let rest while preparing green beans.
Add 1 pound trimmed green beans, remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper to now-empty sheet and toss in rendered chicken juices on sheet. Spread green beans into even layer over sheet, then roast until tender, 12 to 15 minutes, stirring halfway through roasting.
Carve chicken. Slit each potato lengthwise. Using clean dish towel, hold ends and squeeze slightly to push flesh up and out. Sprinkle with 1½ teaspoons grated lemon zest and 1 teaspoon minced fresh parsley. Toss green beans with 1 tablespoon lemon juice and sprinkle with ¼ cup toasted sliced almonds. Serve chicken with potatoes and green beans.