Preheat the oven to 350-degrees F.
Add the onion to the bottom of the roasting pan.
Rinse the chicken, discard the gibbets, and truss the legs. Set the chicken on top of a rack in the roasting pan.
Place the whole cloves of garlic and half the lemon inside the cavity of the chicken.
Massage 2 tablespoons of the softened cowboy butter all over the chicken, including under the skin.
Arrange the potatoes and carrots around the chicken on the roasting pan.
Melt an additional 2 tablespoon of cowboy butter and drizzle over the vegetables.
Cook at 350-degrees F for 20 minutes per pound, or until the internal temperature reaches 165-degrees F (there will be some carryover cooking after you remove it from the oven). Remove from the oven and cover with foil. Let set for 10 minutes.