In a medium mixing bowl, combine the egg yolk, lemon juice, mustard, salt, and sugar whisk until smooth. Continue actively whisking, start slowly adding the oil in small drops. Gradually adding oil is critical for proper emulsification.

Don't stop whisking until about 1/3 of the oil has been added. You'll see that the mixture will become smooth and creamy. Then pause and scrape the sides of the bowl.

Continue whisking the mayo, slowly adding the rest of the oil. Then mix in the grated horseradish and adjust the seasoning to taste.