Roast chicken

Make the butter mixture by mixing together the butter, shallot, lemon zest, etc.
Gently lift the skin of the chicken away from the meat. You can just work your fingers under there. Try not to break the skin.
Smear the butter mixture underneath the skin of the bird, all over. Just work it under there with your fingers. It doesn't have to be anywhere close to perfect.

Position your oven rack 8 inches from the top coil and turn the broiler to high.
When broiler is hot, cook the chicken for about 10 minutes. You are looking for a skin that is dark mahogany and looks very delicious. If it's not there yet, just keep checking it until it's beautiful.
Throughout the cooking process, feel free to maybe rotate the roasting pan a few times to counteract oven hot-spots.
Flip the bird over so that it's skin side down, and continue to cook until the juices run clear when you puncture the bird in the fat part of the thigh, probably about 12-15 minutes. If you like thermometers, the breast is done at 160° and the thigh is done at 165°. I'd probably take it out of the oven a little before it reached those temperatures -- it'll continue to cook a bit after you take it out of the oven, and overcooked chicken sucks.
Remove the roasting pan from the oven, remove the chicken, place it on a serving platter, cover it with foil, and set it aside.

Drain any fat from the roasting pan.
Place the roasting pan on your stove over high heat.
Add the white wine and the sprig of thyme. Let it come to a boil, scraping down the bottom of the pan, until reduced by half.
Add the chicken stock. Boil until reduced by half.
Pour the jus through a strainer. Discard the vegetables and stuff -- they've given up their flavor.

To serve, cut the chicken into quarters. First, cut the chicken half by cutting straight down the middle (there are no bones there since you spatchcocked it). Then cut the thigh/leg quarters away from the breast/wing quarters, making a crescent-shaped cut through the thigh joint. Pour a little jus over each piece, put the rest of the jus in a gravy boat, and serve forth.

How to spatchcock a chicken A much cooler word for "butterflying", spatchcocking allows your chicken to lay open and flat, so that the thickness across the entire bird is pretty much uniform. You won't have to worry about the dark meat undercooking and the breast meat overcooking. It's a skill that you're likely to master on the first try, and a technique you'll use every time you cook a chicken. You will need a pair of poultry shears and a sharp knife (a boning knife is best). If you don't have poultry shears, go buy some. They're only about $8.99 and you'll end up using them all the time in the kitchen -- not just for poultry. Place the chicken on your work surface, breast side down. Using your poultry shears, cut along each side of the backbone. I cut along the right side, then turn the bird around and do the other side, since I'm right handed. Save the backbone for making homemade chicken stock, or discard it. Flip the chicken over and, grabbing it with both hands, open it like a book. Then really bend each side back, like you're trying to ruin the book's binding. You will now remove the keel bone (sternum) of the chicken. Using your sharp knife, cut away any membranes that might be hiding the bone -- the bone is right in the center of the chicken, between the breasts. Your goal is just to work your index finger under the sorta spoon-shaped gray bone and pull it out. There may be some hard white cartilage getting in the way. If there is, cut it out -- you don't want to eat that crap anyway. Just wedge your finger on the bone, and it should come out easily. If there's more of that hard white cartilage near the tail-end of the chicken, cut it out carefully with your sharp knife. Again, it's not especially tasty. Congratulations! You just spatchcocked a chicken!

Brining the chicken Brining the chicken yields a tender and moist finished product, and I can't recommend it enough. In a container large enough to hold two quarts of water plus the spatchcocked chicken (which you may wad up), mix two quarts of water and 12 tablespoons Morton's kosher salt (or ½ cup table salt, or 1 cup Diamond Crystal kosher salt). Put the lid on your container and shake it vigorously. The finished product should pretty much look like water, with nothing sinking to the bottom. You just made brine. Put the chicken in the container with the brine and put it in the refrigerator for three hours. Remove the chicken, rinse it off with clean running water, and pat it dry.