How to Make It
Squeeze the juice from 1 lemon half into a large bowl of water. Working with 1 baby artichoke at a time, pull off and discard the dark green outer leaves. Cut off the top 1/2 inch of the artichoke, then peel and trim the bottom and stem. Halve the artichoke lengthwise and scoop out the furry choke with a teaspoon. Rub with the remaining lemon half and add to the bowl of lemon water.
Wrap the parsley, tarragon, bay leaf and peppercorns cheesecloth and tie into a bundle. In a large saucepan, combine the wine, garlic and herb bundle and bring to a boil. Add the stock, water and 2 teaspoons of salt and return to a boil. Drain the artichokes and add them to the saucepan along with the tomato, carrot, celery, leek and onion. Cover and braise over moderately low heat until the artichokes are tender, about 20 minutes. Let cool completely in the liquid.
Prepare an ice bath. In a large pot of salted boiling water, blanch the asparagus until crisp-tender, about 2 minutes. Using tongs, transfer the asparagus to the ice bath to cool quickly. Drain well.
In a medium bowl, whisk the vinegar with the lemon juice, shallots, chives, parsley, mustard and honey. Gradually whisk in the olive oil. Season the vinaigrette with salt.
Arrange the asparagus on a platter or plates. Using a slotted spoon, arrange the artichokes and some of the braised aromatics on top. Serve the barigoule, passing the vinaigrette at the table.