How to Make It
Fill a large bowl with cold water and squeeze the lemon half into it. Working with 1 artichoke at a time, snap off the outer leaves. Using a sharp knife, trim the stem and base of the artichoke and cut off the top two-thirds of the leaves. With a spoon or melon baller, scrape out the furry choke. Rub the artichoke heart all over with the lemon half and add back to the water. Repeat with the remaining 3 artichokes. Cut 3 of the artichoke hearts into 1-inch pieces and return them to the water.
In a large saucepan, heat the extra-virgin olive oil until shimmering. Drain the artichoke pieces and pat dry. Add them to the saucepan along with the potatoes, shallots and garlic; season with salt. Cook over moderate heat, stirring occasionally, until the vegetables are lightly browned and barely tender, about 20 minutes. Add the wine and cook until nearly evaporated, about 7 minutes. Add the chicken stock, Parmesan rind and thyme bundle and bring to a boil. Cover partially and cook over moderately low heat until the vegetables are tender, about 30 minutes. Discard the thyme bundle and Parmesan rind.
Working in batches, puree the soup until smooth. Return it to the saucepan. Add the heavy cream and season with salt. Keep the soup warm.
Drain the whole artichoke heart and pat thoroughly dry. Thinly slice the heart on a mandoline. In a medium skillet, heat the pure olive oil until shimmering. Add the sliced artichoke heart and fry over high heat, stirring occasionally, until golden, about 1 1/2 minutes. Using a slotted spoon, transfer the fried artichoke slices to paper towels to drain; sprinkle lightly with salt.
Ladle the soup into bowls. Mound some Parmesan shavings in the center, top with the fried artichokes and serve.