Begin by zesting one of the large lemons, making sure to avoid the white pith for a clean taste. Juice the remaining lemons until you have approximately 2½ tablespoons of fresh juice, straining it through a fine-mesh sieve to remove any pulp or seeds.

In a small saucepan, combine 1 cup of heavy cream, ⅓ cup sugar, ¼ teaspoon kosher salt, and the lemon zest. Place over medium-low heat, stirring occasionally for about 5 minutes until the sugar dissolves.

Remove from heat and whisk in the lemon juice and 1 teaspoon of vanilla extract until well combined.

Strain the mixture through a fine-mesh sieve into a clean bowl to ensure a silky texture.

Using a spoon, carefully fill the hollowed lemon halves with the posset mixture.

Cover the filled lemon halves with plastic wrap, and refrigerate them for at least 4 hours to set.

Before serving, let the posset sit at room temperature for 5–10 minutes to enhance the flavors.