Make sure you have a container and lid large enough to fit and submerge the turkey breast. Unwrap the turkey and remove any organs from inside the neck or breast cavity. Pat it dry with paper towels and let it come to room temperature. Wash your hands.
Soak the wood chips in water and set aside.
Heat a quart of water in a pot and whisk in the salt, honey and brown sugar. Bring the mixture to a boil and make sure to stir the salt and sugar until they dissolve. Turn the heat off and add the rest of the brine ingredients. Let the spices steep for 10 minutes as it cools.
Add the brine and the turkey to your container. Add enough ice to cover the turkey. Cover and store in the fridge for up to two days. Check that the meat is submerged, add cold water if the ice wasn't enough.
While you wait make the butter. Leave a stick of butter on the counter for a couple of hours to soften or if it's rock hard, use a grater to quicken the process. Chop chives finely with the garlic and fold into the butter. If you want a less chunky and bright green butter to go under the turkey skin, pass the butter through a couple pulses of the food processor (but not necessary).
If you have a dedicated spice grinder, pulse the peppercorn and coriander until you don't see whole seeds anymore. I like it coarse but if you prefer you can make it fine. If there be no spice grinder in your kitchen, simply crush the seeds with the handle of a wooden spoon, bash a ziplock bag or even empty your pepper grinder and pass the seeds through--though I understand that might take a while to grind out.
Once the turkey is brined, drain the container and dry the meat with paper towels. Take the butter out of the fridge to soften.
On a cutting board, carefully lift up the turkey's skin and run your hands as best as you can underneath it. Like, get in there but don't rip it. Spread 2 to 4 tablespoons of chive butter under the skin. If you can't get it to distribute evenly, at least prioritize the top because the butter will melt and cascade down. Sprinkle the rub all over the turkey breast and place on a pan that will fit in your grill or smoker. Stuff the fennel bulb into the empty cavity.