I also like to cut down a bit on cooking time by starting the leeks in a cold pan. Since I’m not trying to sear the cut sides, they don’t need to be dropped into hot oil; it works just as well to heat them up with the oil.
My final time-saving trick is to use a relatively wide cooking vessel for the braise. An 11- or 12-inch saute pan with high, straight sides is the best tool for the job, but a large Dutch oven will also work. The increased surface area of these pans means the cooking liquid will come to a boil faster and more of the meat and vegetables will make contact with the hot liquid, which decreases the total cooking time.