By hand: Place egg yolk, mustard, lemon juice, 1/4 teaspoon salt, and a pinch of pepper in a bowl and whisk together until mixture is smooth and thoroughly combined. Add the oil: Start with one drop at a time and whisk constantly until mixture begins to thicken; add remaining teaspoon lemon juice, then pour in oil in a very slow, steady stream and continue whisking until oil is incorporated and sauce is thick and emulsified. Use immediately or refrigerate in an airtight container for up to 1 week.
Blender method: Combine egg yolk, mustard, lemon juice, 1/4 teaspoon salt, and a pinch of pepper in the jar of a blender or in a large glass measuring cup, if using a hand blender. Blend until smooth and thoroughly combined. Add oil, starting with one drop at a time, and blend constantly until mixture begins to thicken. Add remaining lemon juice while continuing to blend, and pour remaining oil in a slow and steady stream until oil is incorporated and mixture is thick and emulsified.