For the salad: Rinse the lemons under warm water, wipe dry and squeeze juice. Place half the juice in a bowl of cold water. Remove the outer leaves from the artichokes. Using kitchen scissors, trim 2-3 cm (approximately 3/4-1 inch) from the top. Cut the artichokes in half lengthwise and remove the fibers inside. Place in the lemon water to prevent browning.
In a saucepan, bring a pot of salted water to a boil with the remaining lemon juice. Add the artichokes and simmer over medium heat until soft, about 12 minutes.
For the dressing: Drain the anchovies and finely chop. Rinse the thyme, shake dry and pluck leaves. In a bowl, whisk the anchovies and thyme with the stock, sour cream, lemon juice and 1 tablespoon of the capers. Use a hand blender to puree. Season with pepper to taste.
Peel the eggs and finely chop. Rinse the parsley, shake dry, pluck the leaves and finely chop.
Remove the artichokes from the water, drain thoroughly and cool slightly. Divide the artichokes between 4 plates. Serve drizzled with the dressing. Serve garnished with the remaining capers, parsley and eggs.