Cut a round of parchment paper the same diameter as the pot you're using and set aside.
In a medium saucepan over medium heat, sweat the carrot, onion and garlic with the fennel seed.
Add the wine, water, vinegar, oil and lemon zest, and season with salt and white pepper.
Bring to a simmer, add the artichokes, cover with the parchment, and simmer slowly until the artichokes are tender.
A good way to tell whether they are done without puncturing them is to remove one from the broth and try to push out the choke with the back of a spoon; it should slip out pretty easily.
Remove the pan from the heat and discard the parchment.
Push the basil in among the artichokes and set aside to infuse and cool.
After the artichokes have cooled enough to handle, press out the chokes with your thumbs; quarter the hearts and return them to the pan.