Add yeast and one tablespoon (15 milliliters) sugar to the bowl of a stand mixer. Pour in warm milk and stir to dissolve. Let stand ten minutes.
Add egg yolks, remaining sugar, salt, butter, and vanilla and mix together on low speed. Continue mixing and gradually pour in the flour. If needed, add more flour until the dough comes together. Cover and let rise one hour or until doubled.
Punch to release the air. Turn dough onto a lightly-floured counter. Roll out to two centimeter thickness. Cut out 7.5 centimeter rounds. Place onto a baking sheet. Let rise one to two hours.
Add lime juice and sugar to a saucepan over medium heat. Stir and cook until sugar has dissolved and juice has started to reduce. Remove from heat. Mix in lime zest. In a separate bowl, whisk together egg yolks. Add the lime juice mixture into the egg yolks a spoonful at a time, whisking to make sure the eggs come up to temperature gradually. Pour the mixture back into the saucepan and cook over moderate heat, stirring, until it reaches the consistency of pudding. Remove from heat and strain through a sieve into a bowl. Stir in butter a cube at a time until smooth and uniform. Set aside to cool.
Cooking the donuts:
Add oil to a deep-sided pot. Heat oil to 350 F (175 C). Add donuts three at a time – do not overcrowd the pot. Fry for five minutes, turning every minute. Remove to paper towel-lined plate. Let cool.
Fill a piping bag with the lime curd. Push the tip of the piping bag into the side of the donut to make a hole, then squeeze in filling.
Melt chocolate in a bain marie. Dip the top of the donut in the melted chocolate. Turn back to be right-side-up and place on a tray. Let chocolate dry.