Bring a saucepan of water to a boil. Prepare a bowl of ice water and have it standing nearby.
Place basil leaves into boiling water and blanch until leaves are bright green and slightly soft, about 10 seconds.
Remove basil leaves with a fine mesh strainer and place into ice water to stop the cooking process.
Drain basil, wrap in a paper towel, and gently squeeze to remove excess moisture.
Place egg yolks into a container with a base slightly larger than the base of a stick blender, such as a measuring cup or large beer glass.
Add garlic, lemon juice, salt, cayenne pepper, black pepper, basil leaves, and Dijon mustard to egg yolks.
Pour vegetable oil over the top.
Push stick blender all the way to the bottom of the container. Pulse blender 6 or 8 times, about 1 second per time, until mixture begins to emulsify on the bottom. Once mayonnaise begins to form, raise the stick blender slightly in the container and continue to blend until all oil has run through the blender, mayonnaise is thick, and basil is finely chopped, 10 to 20 more seconds. Taste and adjust seasonings.
Transfer to a bowl. Mayonnaise will keep, refrigerated, for 2 to 3 days.
