One to two days before roasting, mix the kosher salt, garlic salt, brown sugar, and black pepper. Pat the turkey dry and rub the mixture all over, getting it inside the cavity and under the skin wherever you can reach. Set the turkey on a rack over a sheet pan, uncovered, and refrigerate for 24 to 48 hours.
Take the turkey out of the fridge about an hour before you want to roast so it comes up in temperature. Preheat the oven to 325°F. Do not rinse the turkey, just pat it dry with paper towels.
Mince the rosemary, thyme, and sage, then mash them into the softened butter. Gently work your fingers under the skin over the breast and thighs to loosen it, then push about two-thirds of the herb butter underneath, spreading it as evenly as you can. If you'd rather inject, melt the herb butter and inject it under the skin instead, or do both. Either way, the butter needs to get under the skin, and don't be shy with it. Rub the rest all over the outside of the bird.
If you're stuffing the bird, fill the cavity loosely with prepared stuffing. Don't pack it in too tight, it needs room for hot air to circulate. Tuck the fresh herb sprigs under the wings, tie the legs together with kitchen twine, and tuck the wing tips under the body so nothing burns.
Scatter the onions, carrots, and celery across the bottom of a roasting pan and pour in the chicken broth to coat the bottom. Set the turkey on top of the vegetables.
Soak the cheesecloth in the melted butter until it's saturated, then drape it over the breast and legs. Put the whole pan in the oven.
Roast until done, using the cheesecloth as your auto-baster. You don't need to baste while it's on. Figure roughly 13 to 15 minutes per pound at 325°F for a stuffed bird, a little less if unstuffed, but go by temperature, not the clock. About 45 minutes before you think it'll be done, pull the cheesecloth off so the skin can brown and crisp up. From there, baste with the pan juices every 20 minutes or so if you like.
The turkey is done when a thermometer in the thickest part of the thigh reads 165°F, and any stuffing in the cavity should also hit 165°F. Let it rest for 30 minutes before carving so the juices settle.
For the gravy, pour the pan drippings through a strainer into a measuring cup and let the fat rise to the top. Skim off most of the fat, leaving about 1/4 cup in the pan along with all the flavorful juices. Set the roasting pan over medium heat (or transfer to a saucepan) and whisk in the flour to make a roux. Cook for a minute or two, stirring, until it smells toasty. Slowly whisk in the chicken stock, a little at a time, until smooth. Bring to a simmer and let it thicken, adding more stock if you want it thinner. Season with salt and pepper to taste.
