Mix the salt, sugar, and 1 tablespoon of mustard powder in a bowl. Evenly coat the pork with the dry brine, and refrigerate uncovered for at least 5 hours, but preferably overnight.
Remove the pork from refrigerator, and let it sit for 20 minutes at room temperature. Preheat the oven to 300 F. In a large Dutch oven, heat the oil over medium heat. Add the garlic, cut-side down, until golden brown. Set aside.
Add the pork to the pot, and brown on all sides, about 2 minutes per side. Set pork aside, and remove the oil from pot with a paper towel.
Add the stock, miso, soy sauce, and remaining mustard powder to pot, and whisk to combine. Bring to a simmer, and add garlic, sliced ginger, and browned pork.
Cover with a lid, and place it in the oven. Cook for 3 hours, then uncover, and cook until tender and the sauce has reduced, about 2 more hours.
Remove the pork from liquid, and set aside. Strain the braising liquid through a fine-meshed sieve, making sure to squeeze the garlic cloves out the bulb, pressing the paste though the sieve. Skim the fat from the sauce with a spoon.
Spoon the sauce over pork. Lightly shred meat, and spoon more sauce over top. Sprinkle with sesame seeds, if using.