In a blender, combine milk and ground flax seeds. Blend on high speed until the ground flax is barely noticeable and the mixture is frothy, about 1 minute.
Add the sugar, dry mustard, onion powder, salt, vinegar, and lemon juice and blend for a few seconds to combine.
Now begin to add the oil. With the blender running, use the hole in the lid to stream in 1 tablespoon at a time, blending for about 30 seconds after each addition (if using a high-speed blender like Vitamix, 5 to 10 seconds should do it). Give your blender a break every now and then so that it doesn’t heat up the mayo. You should notice it thickening after about half the oil has been added. By the time you’ve used three-fourths of the oil, it should be spreadable. And with the last addition, you should have a thick mayo. If it seems watery, keep blending.
Straight out of the blender, the mayo will probably taste saltier and tangier than you’d like, but the flavors mellow as it sets. Transfer to a glass container, seal tightly, and refrigerate for a few hours. The mayo will thicken even further. Use within a week.