Preheat your oven to 340°F/170°C and line a mini muffin pan with parchment paper or aluminum foil and place an eggshell into each well.
In a large bowl, combine the sugar, cake flour, baking powder, baking soda, and salt. Add the softened butter until the mixture resembles fine sand. Set aside.
In a separate bowl, combine the eggs, greek yogurt, olive oil, and vanilla extract. Pour this mixture into the dry ingredients and mix until just combined.
Slowly add the milk and stir.
Fill the eggshells halfway and bake for 16~18 minutes. If you insert a toothpick and it comes out clean, they're done. Transfer them onto a wire rack immediately and let them cool completely. Clean off any cake that spilled over using a knife to scrape and then a wet paper towel.
Using a toothpick, wiggle a hole into each cake.
Put the carrot curd into a piping bag with a small round piping tip and pipe the curd into each cake.