Step 1

In a medium heat-safe bowl, whisk the egg yolk, vinegar, and 1 tablespoon of water.

Step 2

Fill a large bowl with ice water and set it aside.

Step 3

In a saucepan over medium-low heat, bring 2 inches of water to a simmer.

Step 4

Carefully set the bowl with the yolk mixture over the saucepan. Do not allow the bowl to touch the water.

Step 5

Slowly whisk the yolk mixture until an instant thermometer inserted in the center reads 145 degrees F, about 4-5 minutes.

Step 6

Set the bowl with the cooked yolk mixture in the ice water to stop the cooking.

Step 7

Whisk the cooked yolk mixture just until it is lukewarm. Do not allow it to get cold.

Step 8

Place the bowl with the lukewarm yolk mixture on a damp towel.

Step 9

Add 1/4 teaspoon of salt to the yolk mixture.

Step 10

In a measuring cup, combine the vegetable oil and the extra-virgin olive oil.

Step 11

Whisking the yolk mixture constantly, slowly drizzle the oil mixture into the bowl in a thin stream until the yolk mixture is smooth and thick. If at any point the oil mixture is not emulsifying into the mixture, stop adding the oil and whisk to combine before continuing. If the mayo is too thick, whisk in 1-2 teaspoons of water.

Step 12

Season the mayo with salt, to taste.

Step 13

The leftover mayo can be covered and chilled for up to 4 days.

Step 14

Spread the mayo on 1 piece of bread.

Step 15

Top the mayo-covered bread with lettuce, tomato, ham, pickles, and cheese.

Step 16

Spread the mustard over the remaining piece of bread.

Step 17

Close the sandwich.

Step 18

Serve.