Whisk the egg yolk, vinegar and 1 tablespoon water in a medium nonreactive heatproof bowl (glass or ceramic). Fill a large bowl with ice water and set aside. Bring 2 inches water to a simmer in a saucepan over medium-low heat.
Set the bowl with the yolk mixture over the saucepan (do not let the bowl touch the water). Slowly whisk until a thermometer inserted into the yolk mixture registers 150 degrees F, 4 to 5 minutes.
Set the bowl with the yolk mixture in the ice water to stop the cooking; whisk just until the mixture is lukewarm (do not let it get cold).
Put the bowl with the yolk mixture on a damp towel. Add 1/4 teaspoon salt. Combine the oils in a measuring cup. Whisking constantly, slowly drizzle the oil into the bowl in a thin stream until smooth and thick. If at any point the oil is not emulsifying into the mixture, stop adding the oil and whisk to combine before continuing. Whisk in 1 to 2 teaspoons water if the mayonnaise is too thick. Season with salt. Cover and refrigerate up to 4 days.
This homemade mayonnaise is safe: We used a pasteurized egg and heated the yolk to kill any bacteria.