How to Make It
In a small saucepan, cover the potatoes with water. Add a large pinch of salt and bring to a boil. Simmer the potatoes over moderately high heat until just tender when pierced with a knife, about 8 minutes. Drain and let cool. Peel and cut the potatoes into 1/2-inch-thick wedges.
Working with one artichoke at a time, pull off and discard the dark green outer leaves. Using a small, sharp knife, trim off about 1/2 inch of the leaves and peel and trim the stem. Quarter the artichoke and scrape out the hairy choke. Rub the artichoke all over with lemon and squeeze the extra lemon juice into a small bowl of water. Add the quartered artichoke to the lemon water. Repeat with the remaining artichokes.
In a small skillet, melt 1 tablespoon of the butter. Drain the artichokes. Add them to the skillet and cook over moderate heat, stirring a few times, until golden brown, about 2 minutes. Add the white wine and bring to a boil. Add the water, cover and cook over moderate heat, stirring occasionally, until the artichokes are just tender, 12 minutes.
In a large skillet, melt 1 tablespoon of the butter. Add the mushrooms and season with salt. Cover and cook over moderate heat, stirring occasionally, until browned and tender, about 4 minutes. Push the mushrooms to one side of the skillet and add the remaining 2 tablespoons of butter. Add the shallot and cook until softened, about 4 minutes. Add the potatoes, season with salt and turn to coat with the butter. Cook until the potatoes are golden brown, about 2 minutes per side. Gently fold in the mushrooms, artichokes and their cooking liquid. Season with salt and transfer to a shallow serving bowl. Garnish with the cheese shavings and serve.
The whole cooked potatoes and artichoke quarters can be refrigerated separately overnight. Gently reheat the artichokes and bring the potatoes to room temperature before proceeding.