Step 1Place the chicken in a medium bowl and sprinkle with one-half teaspoon of paprika, one-half teaspoon salt, a few grindings of pepper and half the garlic. Toss the chicken to coat with the mixture, then set aside.
Step 2Bring the chicken broth to a simmer in a large saucepan and add the saffron, crushing it with your fingers as it goes in. Turn the heat to low and cover until ready to use.
Step 3Fill a medium saucepan with water and bring it to a boil. Add the shelled fava beans, blanch for 2 minutes, then drain and shock in ice water. Peel the skin from each fava bean; discard the skins. Set aside.
Step 4Cut the tomatoes in half horizontally and grate them on a box grater into a large bowl, discarding the skins. Set aside.
Step 5Fill a medium bowl with cold water, squeeze the lemon halves into it, and drop in the squeezed lemons. To trim an artichoke, pull off all the outer leaves until only pale, tender leaves remain. Trim the bottom of the stem and peel the stem. Use a sharp knife to cut off the tips, then cut it in half vertically (if the artichoke is very small) or in quarters (if it's bigger). Drop the pieces into the lemon water. Trim the rest in the same manner. Leave them in the acidulated water until ready to use.
Step 6Heat 3 tablespoons of olive oil in a 16-inch paella pan over high heat until it's smoking. Add the chicken, and saute for about 3 minutes over high heat, until the pieces start to brown. While they're browning, drain the artichokes, pat them dry and add them to the chicken; continue sauteing for about 7 to 8 minutes until the artichokes are tender and beginning to caramelize (you may need to turn down the heat a bit). Add the string beans and continue sauteing for about 2 minutes. Add the red pepper and a good pinch of salt, and stir to combine. Push everything to the edges of the pan.
Step 7Add the remaining olive oil to the center of the pan. Lower the heat to medium, add the remaining garlic, and cook until fragrant, about 15 seconds. Add the tomato puree to the center of the pan and cook, stirring occasionally, until the tomatoes are thickened, about 5 to 7 minutes. Push the chicken and vegetables into the tomato mixture, and stir to combine. Add the remaining paprika, the sprig of rosemary, the cayenne, and a good pinch of salt and stir. Add the rice and stir to combine.
Step 8Meanwhile, heat the oven to 425 degrees or light a grill with a lid. If using a grill, do not continue with the recipe until the grill is nearly ready, when the hot coals are coated with white ash.
Step 9Add 4 cups of chicken broth to the paella pan, and stir to combine. Shake the pan gently to distribute the rice evenly. Move the pan over two burners and continue to cook over medium heat until the cooking liquid is about level with the rice but the rice is still soupy, about 7 minutes. Periodically move and rotate the pan so that the liquid boils evenly. If the liquid is absorbed too quickly, sprinkle on some additional broth.
Step 10If using a grill, when the rice is almost done, distribute the coals evenly on the bottom grate. Transfer the pan to the grill and close the lid, or to the oven. Bake or grill for about 15 minutes, until the rice is tender, but a little al dente. Check on it a couple times as it cooks, and if seems too al dente, sprinkle on a little more broth.
Step 11Remove the paella from the oven or grill, cover the pan with foil and let it stand for 5 minutes. Uncover it and let it stand an additional 10 minutes. Serve from the pan.