Preheat the oven to 425°F.
Line a baking sheet with parchment paper.
Place the whole chicken breast-side down on a sturdy cutting board.
Using sharp kitchen shears, cut along both sides of the backbone.
Remove the backbone completely.
Open the chicken and gently spread the rib cage.
Using a heavy sharp knife, score down the length of the sternum.
Turn the chicken over so the breast side faces up.
Press firmly on the breastbone with the palm of your hand until the chicken lies flat.
Season the inside of the chicken with 1/2 tsp salt and 1/8 tsp black pepper.
Transfer the flattened chicken to the prepared baking sheet, breast-side up.
Carefully loosen the skin from the chicken meat by sliding your fingers between the skin and the meat.
Loosen the skin over the breast and leg areas, taking care not to tear it.
In a small bowl, combine the softened butter, olive oil, parsley, minced garlic, lemon zest, lemon juice, salt, and black pepper.
Mash with a fork until the compound butter is well blended.
Spread approximately 2/3 of the compound butter under the loosened chicken skin.
Distribute it evenly over the breast and leg meat.
Spread the remaining 1/3 of the compound butter over the outside of the chicken skin.
Arrange the quartered red potatoes, quartered carrots, and halved Brussels sprouts around the chicken on the baking sheet.
Drizzle the vegetables lightly with olive oil.
Season the vegetables with salt and black pepper to taste.
Roast for approximately 45 minutes, or until a meat thermometer inserted into the thickest part of the breast and thigh reaches 165°F.
Remove the chicken from the oven.
Allow the chicken to rest uncovered for 10 to 15 minutes before slicing.
Serve warm with the roasted vegetables.