Place the egg, garlic and lemon juice in the blender cup or jar that just fits the head of your immersion blender. Add the vegetable oil and olive oil and allow to settle for 15 seconds.
Place head of immersion blender at the bottom of cup and turn it on high speed. Do not move the head for 30 seconds or until a white, creamy, thick mayo forms at the bottom by the head.
As mayonnaise forms, slowly tilt and lift the head of the immersion blender until all oil is emulsified. Store in an airtight container in the refrigerator for up to 10 days.