Heat ¼ cup oil in a deep-sided saute pan with the onion, the whole garlic bulb, fennel seed, and bay leaves over medium heat. Season with salt and pepper and cook until onions are tender, about 8 minutes. Remove from heat while you prepare the artichokes (see below: “How to Prepare Artichokes”).
When the artichokes are trimmed, place them in the pan with the onion mixture. Add water to cover and season generously with salt and pepper. Add the lemons to the pan, cover, and cook over medium-low heat until tender when pierced with a skewer, 20 to 30 minutes.
In a separate pan, heat 2 tablespoons olive oil over medium-high heat. Add the leeks, season with salt and pepper and cook until tender and golden, about 10 minutes. Transfer the leeks to a plate and wipe out the pan with a paper towel. Add the remaining 2 tablespoons oil to the pan and raise heat to high. Add half the mushrooms, season with salt and pepper and cook until tender, about 5 minutes. Repeat with remaining mushrooms, then transfer to a plate with the leeks.
Meanwhile, when artichokes are done, transfer to a plate, discarding the lemons. Squeeze the cooked garlic into the pan, discarding the skin. Bring liquid to a boil and cook over high heat about 10 minutes, or until liquid has reduced to about ¼ cup. Set aside. To serve, bring everything to room temperature. Combine artichokes, leeks, and mushrooms in a bowl. Toss well. Add reserved onion-garlic mixture, prosciutto, and chopped thyme. Toss well; season with salt and pepper if necessary.
How to Prepare Artichokes
Cut lemons in half; set aside. Lay the first artichoke on its side and cut across the top 1 or 2 inches (the top quarter of the artichoke is mostly prickly leaf ends). Rub the trimmed leaves with lemon. Gently snap or cut back the tough upper portion of each outer leaf, leaving the meaty part of the leaf still attached. Continue working down the artichoke until remaining leaves are pale green and tender. Trim the tough skin around the base of the artichoke. Pare the skin off the stem and trim the tip. Rub the entire artichoke with lemon. Cut each artichoke in half lengthwise and remove the choke, the thistle-like prickly center. Rub the halves completely with lemon.