Preheat oven to 450F and move the oven rack to the lower middle position.
Fill a large bowl with cold water. Add a few tablespoons of lemon juice. Take each baby artichoke and cut about 1/3 off the top. Then pull off the leaves until you are left with only the very light green to yellow tender inner leaves. Trim off the stem and cut the artichoke into quarters. As you finish each artichoke, put it in the lemon water.
Take a large, straight sided, oven safe saute pan and heat it up over medium high heat with the olive oil. Once the pan is hot and the oil is shimmery, sprinkle the chicken thighs with kosher salt and black pepper and put them, skin side down, in the pan.
Sear the chicken, skin side down, for 5 minutes then turn them and sear for another 3 minutes on the other side. Transfer the chicken to a bowl.
Drain out most of the fat in the pan, leaving about 2 tablespoons. Drain the artichokes and pat them as dry as you can. Add the artichokes to the pan along with the shallots. Cook the vegetables for 5-7 minutes stirring frequently, or until they are a little browned on each side.
Add the flour to the pan and stir it in well. Cook for 1 minute.
Add the metaxa (or brandy) to the pan. Once it is almost cooked off, about 10 seconds, add the stock and lemon juice. Bring to a simmer. Add back the chicken, skin side up, along with the garlic and dill.
Put the chicken in the oven and roast for 12-15 minutes, or until the chicken is cooked through and reaches about 160F on a kitchen thermometer.