Hold the immersion blender at the bottom of the jar and pulse on and off 10-20 times then hold the power on.
Move the blender up and down slowly until ingredients are well blended and emulsified, about 30-60 seconds.
Add buttermilk and cheese to the jar and blend until combined.
1/4 cup buttermilk, 3/4 cup grated parmesan cheese
Store in the refrigerator for up to a week.
You can also use a blender:
Put all ingredients except the olive oil, milk, and parmesan cheese into the blender container.
Turn machine on (I use a medium low speed on my Vitamix) and s l o w l y pour the oil in through the feed tube to emulsify the dressing. Adding the oil slowly is the key to emulsification. It should take about 8-10 minutes to add the oil.
Stir in milk and cheese.
Store in the refrigerator for up to a week.
Notes
It's pretty critical that the egg be at room temperature. 75 degrees or so is ideal. If your house is cooler than that, let the egg sit in a bowl of warm water for a little while.
If the dressing thickens too much after refrigeration, add a few tablespoons of milk to thin it out.