Add the chilled milk, Dijon mustard, salt, and sugar to a tall, narrow blending container.
Pour the neutral oil over the milk mixture and allow everything to settle for about 30 seconds.
Place an immersion blender firmly against the bottom of the container and blend without moving it until a thick white emulsion begins forming at the bottom.
Slowly lift and lower the blender through the mixture until all of the oil is incorporated and the mayonnaise is thick and creamy.
Blend in the lemon juice and white vinegar just until smooth. Refrigerate for at least 30 minutes before serving so the mayonnaise becomes slightly thicker and the flavor rounds out.