Fill a large bowl with water and add the leeks, swish around with your hands to get the sand out from in between the layers. In a large saute pan over medium heat, add 1 tablespoon each of butter and olive oil. Add the leeks, sprinkle with the thyme, salt and pepper and cook for about 10 minutes, turning every now and then, until the leeks are softened but not colored. Remove the leeks from heat to cool.
In a large bowl combine the leeks, cornbread, and chicken stock and season with salt and pepper.
Use a sharp, thin knife to cut down the length of the turkey cutting the breast parallel to the board, almost in half just stopping short of the outside edge. Open out the 2 halves as if you were opening a book. Now you've got a large piece of meat that will cook evenly and is thin enough to roll. Place 2 cups of the stuffing inside each of the breasts and then fold the turkey back over.
Place the remaining stuffing in a small baking dish and cook alongside the turkey.
Tie in 4 places with kitchen twine and season with salt and pepper.
Heat the remaining tablespoon of butter and olive oil in a roasting pan over medium-high heat. Put the turkey in the roasting pan and sear all over. Transfer the pan to the oven and roast for about 25 minutes (internal temperature should reach 160 degrees F on an instant-read thermometer). Take the turkey out of the oven and let it rest for 5 to 10 minutes, then cut crosswise into slices.