It is very important that your egg is at room temperature. Place in a bowl of hot water to short cut this step.
Put egg and lemon juice in food processor with mustard and salt.
Turn machine on and pour the 1/4 cup olive oil down the feed tube. Process for 1 minute.
Now, slowly drizzle in the remaining 1 cup olive oil, pouring it down the feed tube in a very thin stream. At first it will splatter around but just keep drizzling and soon it will smooth out.
Keep drizzling, and drizzling, about 4 minutes.
When it is all incorporated run machine for another 30 seconds. Makes about 1 1/4 cups