In a large dutch oven, brown chicken wings in a couple of tablespoons of vegetable oil on both sides, working in batches. When they're all done, return them to the dutch oven along with any accumulated juices.
Add minced garlic and stir, turning heat down to medium-low.
In a large measuring cup, combine enough of the beer with the brown sugar to completely dissolve the sugar.
Add the dry mustard, soy sauce, and white vinegar to the brown sugar/beer mixture and stir to combine. Pour over the chicken, stir, then add the remaining beer.
Turn the heat up until the sauce begins to simmer. Stir frequently to avoid scorching, turning down if sauce boils. Let cook until sauce gets nice and thick and nearly caramelizes, which can take up to an hour.
Sprinkle with sesame seeds and stir to coat the wings.
Remove from heat and drizzle with dark sesame oil. Do not cook the sesame oil.