Add the egg yolk, water, lemon juice, kosher salt, and cayenne pepper into a cup that just fits the head of an immersion blender. I use a 16-ounce coffee mug. You can also use the cup that comes with the blender if it's narrow enough.
Cube the butter and place it in a glass measuring cup. Cover and melt the butter in the microwave in 30-second increments. The butter should be hot, not just melted - see notes below.
Place an immersion blender into the bottom of the cup and turn it on. With the blender running, very slowly pour the hot butter into the cup. The butter will emulsify with the egg yolk and lemon juice, resulting in a thick, creamy sauce.
Transfer the sauce to a serving dish. Serve immediately. This sauce cannot be cooled and reheated, but it will keep at room temperature for up to an hour.
The CDC recommends cooking eggs until both the yolk and white are firm.
You want the butter to be hot, not just melted. If you worry about it splattering in the microwave, heat it in a small saucepan on the stovetop over medium-high heat until it's melted and the foam subsides.
To further increase your chances of success and reduce the risk of the sauce separating, use ghee (clarified butter) instead of butter. Clarified butter contains no water, making it easier to stabilize and thicken the sauce.
For the same reason, you can initially omit the water. Add it later if the sauce is too thick.
Unfortunately, you can't store the leftovers. This sauce cannot be refrigerated and reheated, so try to make only as much as you can use right away.