Preheat your oven to 350°F (175°C). This ensures that your cookie cups bake evenly.
In a large mixing bowl, combine the softened butter and granulated sugar. Using an electric mixer or a whisk, beat them together until the mixture is light and fluffy, about 2-3 minutes.
Add the large egg and vanilla extract to the butter-sugar mixture. Beat until fully incorporated.
In a separate bowl, whisk together the baking soda, salt, cream of tartar, and all-purpose flour. Gradually add this dry mixture to the wet ingredients, mixing until just combined. Be careful not to overmix.
Scoop about 1.5 tablespoons of dough into each cup of your muffin tin. Use your fingers or the back of a spoon to press the dough into the bottom and up the sides, creating a small cup shape.
Bake in the preheated oven for 12-15 minutes, or until the edges are lightly golden. Keep an eye on them to avoid over-baking.
Once baked, remove the cookie cups from the oven and let them cool in the tin for about 5 minutes. Carefully transfer them to a cooling rack. Once cooled, fill each cookie cup with lime curd, using a spoon or piping bag for neatness.