Remove small, random little leaves that stick out from the top of the artichokes' stems.
Turn the artichokes stem side down under a running faucet and let the water fill the spaces between the leaves. Turn it back over to drain, then repeat.
Chop off 1/4-inch from the end of the stems to remove any dry and darkened spots.
Place the artichokes in a pot of water with a teaspoon of salt.
Cover and bring to a rolling boil over medium-high heat, then lower the flame to medium. If your lid does not have a venting hole, place the lid at an angle to allow some steam to escape.
Cook until one of the larger leaves on the top easily releases from the stem with a firm tug. That will be approximately 25 minutes after you achieve a rolling boil.
Remove from the pot and drain.
Stand your artichoke on the leaves with the stem up. Using a sharp knife, cut through the center of the stem and down through the leaves to make 2 equally sized halves.
Then remove the small, paper-thin leaves in the center and scrape out the chokes.
Lightly coat the cut side of the artichoke in flour. Just gently place it face down on the flour but do not press it down.
Heat a large, non-stick skillet to medium-high heat. Add one tablespoon of the butter and the olive oil. When the butter has melted, place the artichokes dredged side down and cook until the flour has turned a nice golden brown, approximately 2 minutes.
Place the artichokes browned side up in a large serving platter with raised sides.
De-glaze the pan with the chicken broth and wine, stirring to loosen all the little bits of browned flour.
Then add the garlic, lemon juice, capers, and remaining butter. Taste for seasoning. Add salt and fresh cracked pepper to taste, if necessary.
Cook, stirring occasionally, until reduced by about 1/4. The sauce will still be runny.
Pour the piccata sauce directly out of the pan equally over the artichokes in the serving platter. Be sure to scrape all the bits of garlic and capers out of the pan, then serve.