Place the egg, 1/4 cup of oil, mustard powder, and salt in a mixing bowl, blender, or food processor. Mix thoroughly till thick. I find this is best done in a high speed blender (like Vitamix) or food processor.
While the food processor or blender is running (or while you're whisking), SLOWLY drizzle in the remaining cup of oil in a steady stream. If you dump this in all at once it will be a gloopy disaster. The slower you can do this, the thicker the mayo will be.
Once you've added the oil and the mixture has emulsified, slowly add in the distilled white vinegar to taste. Stir gently to incorporate.