Put about an inch of water in the bottom of a double boiler. Place over high heat to bring to a simmer. Once simmering, turn heat to low to keep the water barely simmering. Prepare an ice bath in a large mixing bowl by filling it about halfway with ice water. Place the ice bath near the stove burner you are using, so it is easily accessible.
Whisk the egg yolk in the top part of the double boiler until smooth. Whisk in the water and the lemon juice.
Place the top part of the double boiler (with egg yolk mixture) over the simmering water. Whisking vigorously, heat to a temperature of 160º Fahrenheit. Use an instant read thermometer for this step.
Once it reaches 160º, remove it from heat and place the bowl in the ice bath, continuing to whisk until the mixture cools. (Don't get water from the ice bath into the egg yolk mixture.
Whisk the dry mustard, sea salt, and cayenne pepper into the egg yolk mixture. Very gradually add small amounts of the avocado oil to the egg yolk mixture while whisking vigorously. Start with a teaspoon or so at a time, whisk in until absorbed, then add a little more. Adding too much oil at once will cause the mixture to "break."
Store in the refrigerator in an airtight container for up to 3 days.