Prepare a sachet by placing the tea in a bit of cheesecloth and tying it with twine.
In a large stockpot, bring the salt, sugar, and 4 quarts of cold water to a boil, stirring to dissolve.
Remove from the heat, add the tea sachet, and let cool completely. Remove the sachet and discard.
Transfer the brine to a large vessel, place the turkey into the brine, and add enough cool water to fully submerge the turkey. Cover and refrigerate for at least 24 hours and up to 36 hours.
Remove the turkey from the brine and place it, breast side up, on wire rack in a roasting pan. Refrigerate, uncovered, allowing the skin to dry out for at least 12 hours, preferably overnight.
Remove the turkey from the refrigerator and let it come to room temperature, about 2 hours.
Preheat the oven to 350°F.
Stuff the cavity of the turkey with the orange and as many of the onion wedges as will reasonably fit. Gently tuck the wings behind the body and using kitchen twine, tie the legs together to cinch up the cavity.
Add the remaining onion wedges to the bottom of the pan along with the garlic cloves and bay leaves. Add the cider and top with enough water so there is a good 1/4 inch of liquid in the bottom of the pan.
Brush the turkey generously with olive oil and place in the preheated oven.
Roast until an instant-read thermometer inserted into the thickest part of the thigh reaches 165°F and the juices run clear, 3 to 3 1/2 hours, depending on your oven and the size of your bird. Monitor the turkey for perfect coloring, covering the bird with foil if over browning at any point.
Remove from the oven and rest, covered with foil, for at least 20 minutes before carving.
